With the growing demand for high-precision flow field simulations in computational science and engineering,the super-resolution reconstruction of physical fields has attracted considerable research interest.However,tr...With the growing demand for high-precision flow field simulations in computational science and engineering,the super-resolution reconstruction of physical fields has attracted considerable research interest.However,tradi-tional numerical methods often entail high computational costs,involve complex data processing,and struggle to capture fine-scale high-frequency details.To address these challenges,we propose an innovative super-resolution reconstruction framework that integrates a Fourier neural operator(FNO)with an enhanced diffusion model.The framework employs an adaptively weighted FNO to process low-resolution flow field inputs,effectively capturing global dependencies and high-frequency features.Furthermore,a residual-guided diffusion model is introduced to further improve reconstruction performance.This model uses a Markov chain for noise injection in phys-ical fields and integrates a reverse denoising procedure,efficiently solved by an adaptive time-step ordinary differential equation solver,thereby ensuring both stability and computational efficiency.Experimental results demonstrate that the proposed framework significantly outperforms existing methods in terms of accuracy and efficiency,offering a promising solution for fine-grained data reconstruction in scientific simulations.展开更多
Nonlinear science is a fundamental area of physics research that investigates complex dynamical systems which are often characterized by high sensitivity and nonlinear behaviors.Numerical simulations play a pivotal ro...Nonlinear science is a fundamental area of physics research that investigates complex dynamical systems which are often characterized by high sensitivity and nonlinear behaviors.Numerical simulations play a pivotal role in nonlinear science,serving as a critical tool for revealing the underlying principles governing these systems.In addition,they play a crucial role in accelerating progress across various fields,such as climate modeling,weather forecasting,and fluid dynamics.However,their high computational cost limits their application in high-precision or long-duration simulations.In this study,we propose a novel data-driven approach for simulating complex physical systems,particularly turbulent phenomena.Specifically,we develop an efficient surrogate model based on the wavelet neural operator(WNO).Experimental results demonstrate that the enhanced WNO model can accurately simulate small-scale turbulent flows while using lower computational costs.In simulations of complex physical fields,the improved WNO model outperforms established deep learning models,such as U-Net,Res Net,and the Fourier neural operator(FNO),in terms of accuracy.Notably,the improved WNO model exhibits exceptional generalization capabilities,maintaining stable performance across a wide range of initial conditions and high-resolution scenarios without retraining.This study highlights the significant potential of the enhanced WNO model for simulating complex physical systems,providing strong evidence to support the development of more efficient,scalable,and high-precision simulation techniques.展开更多
In oceanic and atmospheric science,finer resolutions have become a prevailing trend in all aspects of development.For high-resolution fluid flow simulations,the computational costs of widely used numerical models incr...In oceanic and atmospheric science,finer resolutions have become a prevailing trend in all aspects of development.For high-resolution fluid flow simulations,the computational costs of widely used numerical models increase significantly with the resolution.Artificial intelligence methods have attracted increasing attention because of their high precision and fast computing speeds compared with traditional numerical model methods.The resolution-independent Fourier neural operator(FNO)presents a promising solution to the still challenging problem of high-resolution fluid flow simulations based on low-resolution data.Accordingly,we assess the potential of FNO for high-resolution fluid flow simulations using the vorticity equation as an example.We assess and compare the performance of FNO in multiple high-resolution tests varying the amounts of data and the evolution durations.When assessed with finer resolution data(even up to number of grid points with 1280×1280),the FNO model,trained at low resolution(number of grid points with 64×64)and with limited data,exhibits a stable overall error and good accuracy.Additionally,our work demonstrates that the FNO model takes less time than the traditional numerical method for high-resolution simulations.This suggests that FNO has the prospect of becoming a cost-effective and highly precise model for high-resolution simulations in the future.Moreover,FNO can make longer high-resolution predictions while training with less data by superimposing vorticity fields from previous time steps as input.A suitable initial learning rate can be set according to the frequency principle,and the time intervals of the dataset need to be adjusted according to the spatial resolution of the input when training the FNO model.Our findings can help optimize FNO for future fluid flow simulations.展开更多
Fourier neural operator(FNO)model is developed for large eddy simulation(LES)of three-dimensional(3D)turbulence.Velocity fields of isotropic turbulence generated by direct numerical simulation(DNS)are used for trainin...Fourier neural operator(FNO)model is developed for large eddy simulation(LES)of three-dimensional(3D)turbulence.Velocity fields of isotropic turbulence generated by direct numerical simulation(DNS)are used for training the FNO model to predict the filtered velocity field at a given time.The input of the FNO model is the filtered velocity fields at the previous several time-nodes with large time lag.In the a posteriori study of LES,the FNO model performs better than the dynamic Smagorinsky model(DSM)and the dynamic mixed model(DMM)in the prediction of the velocity spectrum,probability density functions(PDFs)of vorticity and velocity increments,and the instantaneous flow structures.Moreover,the proposed model can significantly reduce the computational cost,and can be well generalized to LES of turbulence at higher Taylor-Reynolds numbers.展开更多
In this paper,we develop the deep learning-based Fourier neural operator(FNO)approach to find parametric mappings,which are used to approximately display abundant wave structures in the nonlinear Schr?dinger(NLS)equat...In this paper,we develop the deep learning-based Fourier neural operator(FNO)approach to find parametric mappings,which are used to approximately display abundant wave structures in the nonlinear Schr?dinger(NLS)equation,Hirota equation,and NLS equation with the generalized PT-symmetric Scarf-II potentials.Specifically,we analyze the state transitions of different types of solitons(e.g.bright solitons,breathers,peakons,rogons,and periodic waves)appearing in these complex nonlinear wave equations.By checking the absolute errors between the predicted solutions and exact solutions,we can find that the FNO with the Ge Lu activation function can perform well in all cases even though these solution parameters have strong influences on the wave structures.Moreover,we find that the approximation errors via the physics-informed neural networks(PINNs)are similar in magnitude to those of the FNO.However,the FNO can learn the entire family of solutions under a given distribution every time,while the PINNs can only learn some specific solution each time.The results obtained in this paper will be useful for exploring physical mechanisms of soliton excitations in nonlinear wave equations and applying the FNO in other nonlinear wave equations.展开更多
An efficient data-driven approach for predicting steady airfoil flows is proposed based on the Fourier neural operator(FNO),which is a new framework of neural networks.Theoretical reasons and experimental results are ...An efficient data-driven approach for predicting steady airfoil flows is proposed based on the Fourier neural operator(FNO),which is a new framework of neural networks.Theoretical reasons and experimental results are provided to support the necessity and effectiveness of the improvements made to the FNO,which involve using an additional branch neural operator to approximate the contribution of boundary conditions to steady solutions.The proposed approach runs several orders of magnitude faster than the traditional numerical methods.The predictions for flows around airfoils and ellipses demonstrate the superior accuracy and impressive speed of this novel approach.Furthermore,the property of zero-shot super-resolution enables the proposed approach to overcome the limitations of predicting airfoil flows with Cartesian grids,thereby improving the accuracy in the near-wall region.There is no doubt that the unprecedented speed and accuracy in forecasting steady airfoil flows have massive benefits for airfoil design and optimization.展开更多
Traditional aerodynamic optimization coupled with computational fluid dynamics is associated with a high computational cost.Surrogate models based on deep learning methods can rapidly predict flow fields from the grid...Traditional aerodynamic optimization coupled with computational fluid dynamics is associated with a high computational cost.Surrogate models based on deep learning methods can rapidly predict flow fields from the grid input but often suffer from poor accuracy and generalizability.This study introduces a modified Fourier neural operator for flow field prediction.Unlike most convolution-based models,the Fourier neural operator learns the solution operator directly in the function space,enhancing predictive accuracy and generalizability.The proposed model incorporates a shallow feature extractor,a boundary variable finetuner,and several physical priors,including the initial flow field and boundary conditions.The model is trained on uniformly parameterized algebraic grids to accelerate grid generation in aerodynamic optimization.The prediction error for the flow field and force coefficients on the validation and test sets is reduced by 70%to 90%compared with that of the previous convolutional model.The proposed model can make precise predictions for supercritical airfoils under typical working conditions,with a drag coefficient error of approximately 1 drag count on the validation set,and generalizes better than previous convolution-based methods do on extrapolative inflow conditions and airfoils.展开更多
Accurate characterization of temperature-dependent thermoelectric properties(TEPs),such as thermal conductivity and the Seebeck coefficient,is essential for modeling and design of thermoelectric devices.However,nonlin...Accurate characterization of temperature-dependent thermoelectric properties(TEPs),such as thermal conductivity and the Seebeck coefficient,is essential for modeling and design of thermoelectric devices.However,nonlinear temperature dependence and coupled transport behavior make forward simulation and inverse identification challenging under sparse measurements.We present a physics-informed machine learning framework combining physics-informed neural networks(PINN)and neural operators(PINO)for solving forward and inverse problems in thermoelectric systems.PINN enables field reconstruction and property inference by embedding governing equations into the loss function,while PINO generalizes across materials without retraining.Trained on simulated data for 20 p-type materials and tested on 60 unseen materials,PINO accurately infers TEPs using only sparse temperature and voltage data.This framework provides a scalable,dataefficient,and generalizable solution for thermoelectric property identification,facilitating highthroughput screening and inverse design of advanced thermoelectric materials.展开更多
Traditionally,classical numerical schemes have been employed to solve partial differential equations(PDEs)using computational methods.Recently,neural network-based methods have emerged.Despite these advancements,neura...Traditionally,classical numerical schemes have been employed to solve partial differential equations(PDEs)using computational methods.Recently,neural network-based methods have emerged.Despite these advancements,neural networkbased methods,such as physics-informed neural networks(PINNs)and neural operators,exhibit deficiencies in robustness and generalization.To address these issues,numerous studies have integrated classical numerical frameworks with machine learning techniques,incorporating neural networks into parts of traditional numerical methods.In this study,we focus on hyperbolic conservation laws by replacing traditional numerical fluxes with neural operators.To this end,we developed loss functions inspired by established numerical schemes related to conservation laws and approximated numerical fluxes using Fourier neural operators(FNOs).Our experiments demonstrated that our approach combines the strengths of both traditional numerical schemes and FNOs,outperforming standard FNO methods in several respects.For instance,we demonstrate that our method is robust,has resolution invariance,and is feasible as a data-driven method.In particular,our method can make continuous predictions over time and exhibits superior generalization capabilities with out-of-distribution(OOD)samples,which are challenges that existing neural operator methods encounter.展开更多
Digital twins of lithium-ion batteries are increasingly used to enable predictive monitoring,control,and design at system scale.Increasing their capabilities involves improving their physical fidelity while maintainin...Digital twins of lithium-ion batteries are increasingly used to enable predictive monitoring,control,and design at system scale.Increasing their capabilities involves improving their physical fidelity while maintaining sub-millisecond computational speed.In this work,we introduce machine learning surrogates that learn physical dynamics.Specifically,we benchmark three operator-learning surrogates for the Single Particle Model(SPM):Deep Operator Networks(DeepONets),Fourier Neural Operators(FNOs)and a newly proposed parameter-embedded Fourier Neural Operator(PE-FNO),which conditions each spectral layer on particle radius and solid-phase diffusivity.We extend the comparison to classical machine-learning baselines by including U-Nets.Models are trained on simulated trajectories spanning four current families(constant,triangular,pulse-train,and Gaussian-random-field)and a full range of State-of-Charge(SOC)(0%to 100%).DeepONet accurately replicates constant-current behaviour but struggles with more dynamic loads.The basic FNO maintains mesh invariance and keeps concentration errors below 1%,with voltage mean-absolute errors under 1.7mV across all load types.Introducing parameter embedding marginally increases error but enables generalisation to varying radii and diffusivities.PE-FNO executes approximately 200 times faster than a 16-thread SPM solver.Consequently,PE-FNO’s capabilities in inverse tasks are explored in a parameter estimation task with Bayesian optimisation,recovering anode and cathode diffusivities with 1.14%and 8.4%mean absolute percentage error,respectively,and 0.5918 percentage points higher error in comparison with classical methods.These results pave the way for neural operators to meet the accuracy,speed and parametric flexibility demands of real-time battery management,design-of-experiments and large-scale inference.PE-FNO outperforms conventional neural surrogates,offering a practical path towards high-speed and high-fidelity electrochemical digital twins.展开更多
The neural operator theory ofers a promising framework for efficiently solving complex systems governed by partial diferential equations(PDEs for short).However,existing neural operators still face significant challen...The neural operator theory ofers a promising framework for efficiently solving complex systems governed by partial diferential equations(PDEs for short).However,existing neural operators still face significant challenges when applied to spatiotemporal systems that evolve over large time scales,particularly those described by evolution PDEs with time-derivative terms.This paper introduces a novel neural operator designed explicitly for solving evolution equations based on the theory of operator semigroups.The proposed approach is an iterative algorithm where each computational unit,termed the single-step neural operator solver(SSNOS for short),approximates the solution operator for the initial-boundary value problem of semilinear evolution equations over a single time step.The SSNOS consists of both linear and nonlinear components:The linear part approximates the linear operator in the solution map;in contrast,the nonlinear part captures deviations in the solution function caused by the equations nonlinearities.To evaluate the performance of the algorithm,the authors conducted numerical experiments by solving the initial-boundary value problem for a two-dimensional semilinear hyperbolic equation.The experimental results demonstrate that their neural operator can efficiently and accurately approximate the true solution operator.Moreover,the model can achieve a relatively high approximation accuracy with simple pre-training.展开更多
Indoor airflow distribution significantly influences temperature regulation,humidity control,and pollutant dispersion,directly impacting thermal comfort and occupant health.Accurate and efficient prediction of airflow...Indoor airflow distribution significantly influences temperature regulation,humidity control,and pollutant dispersion,directly impacting thermal comfort and occupant health.Accurate and efficient prediction of airflow fields is essential for optimizing ventilation systems and enabling real-time control.However,existing computational approaches for dynamic ventilation are computationally intensive and have limited generalization capabilities.This study leverages the Fourier neural operator(FNO),a method rooted in operator learning and Fourier transform principles,to develop a three-dimensional(3D)airflow simulation model capable of predicting velocity and its components.The model was trained using 200 s of sinusoidal ventilation data(amplitude:0.4)and evaluated under diverse air supply patterns,including sinusoidal(amplitude:0.8),intermittent,and stepwise periodic ventilation.Additionally,the model’s performance was assessed with low-resolution training data and further tested for recursive prediction accuracy.Results reveal that the FNO method achieves high accuracy,with a mean square error of 9.906×10^(-5)for sinusoidal amplitude 0.8 and 4.004×10^(-5)over 400 time steps for sinusoidal,intermittent,and stepwise conditions.Further evaluations,including tests on low-resolution training data and recursive prediction,were conducted to examine the model’s resolution invariance and assess its performance in iterative forecasting.These findings demonstrate the FNO method’s potential for robust,efficient prediction of 3D unsteady airflow fields,providing a pathway for real-time ventilation system optimization.展开更多
Partial differential equations(PDEs)play a dominant role in themathematicalmodeling ofmany complex dynamical processes.Solving these PDEs often requires prohibitively high computational costs,especially when multiple ...Partial differential equations(PDEs)play a dominant role in themathematicalmodeling ofmany complex dynamical processes.Solving these PDEs often requires prohibitively high computational costs,especially when multiple evaluations must be made for different parameters or conditions.After training,neural operators can provide PDEs solutions significantly faster than traditional PDE solvers.In this work,invariance properties and computational complexity of two neural operators are examined for transport PDE of a scalar quantity.Neural operator based on graph kernel network(GKN)operates on graph-structured data to incorporate nonlocal dependencies.Here we propose a modified formulation of GKN to achieve frame invariance.Vector cloud neural network(VCNN)is an alternate neural operator with embedded frame invariance which operates on point cloud data.GKN-based neural operator demonstrates slightly better predictive performance compared to VCNN.However,GKN requires an excessively high computational cost that increases quadratically with the increasing number of discretized objects as compared to a linear increase for VCNN.展开更多
Learning mappings between functions(operators)defined on complex computational domains is a common theoretical challenge in machine learning.Existing operator learning methods mainly focus on regular computational dom...Learning mappings between functions(operators)defined on complex computational domains is a common theoretical challenge in machine learning.Existing operator learning methods mainly focus on regular computational domains,and have many components that rely on Euclidean structural data.However,many real-life operator learning problems involve complex computational domains such as surfaces and solids,which are non-Euclidean and widely referred to as Riemannian manifolds.Here,we report a new concept,neural operator on Riemannian manifolds(NORM),which generalises neural operator from Euclidean spaces to Riemannian manifolds,and can learn the operators defined on complex geometries while preserving the discretisation-independent model structure.NORM shifts the function-to-function mapping to finite-dimensional mapping in the Laplacian eigenfunctions’subspace of geometry,and holds universal approximation property even with only one fundamental block.The theoretical and experimental analyses prove the significant performance of NORM in operator learning and show its potential for many scientific discoveries and engineering applications.展开更多
In this paper,the technique of approximate partition of unity is used to construct a class of neural networks operators with sigmoidal functions.Using the modulus of continuity of function as a metric,the errors of th...In this paper,the technique of approximate partition of unity is used to construct a class of neural networks operators with sigmoidal functions.Using the modulus of continuity of function as a metric,the errors of the operators approximating continuous functions defined on a compact interval are estimated.Furthmore,Bochner-Riesz means operators of double Fourier series are used to construct networks operators for approximating bivariate functions,and the errors of approximation by the operators are estimated.展开更多
An efficient neural mode-solving operator is proposed for evaluating the propagation properties of optical fibers.By incorporating the governing Helmholtz equation into training,the working mechanism of the proposed o...An efficient neural mode-solving operator is proposed for evaluating the propagation properties of optical fibers.By incorporating the governing Helmholtz equation into training,the working mechanism of the proposed operator adheres to the physics essence of fiber analysis.The training of the mode-solving operator adopts a hybrid physics-informed and data-driven approach,providing the advantages of strong physical consistency,enhanced prediction accuracy,and reduced data dependency in comparison with purely datadriven methods.Benefiting from the improvements in network input-output mapping formulation,the proposed operator offers broader applicability to different fiber types and greater flexibility for property optimization.Combined with the particle swarm optimization and refractive index optimization,the operator demonstrates its capacity for the inverse design of multi-step-index fibers(MSIFs)and graded-index fibers(GRIFs).For MSIFs,to ensure a low mode crosstalk for short-distance transmission systems,optimized refractive index profiles(RIPs)of both three-ring and four-ring structures are obtained from large structure parameter search spaces.For GRIFs,to ensure a low receiving complexity for long-haul transmission systems,optimized RIP with low root mean square mode group delay is obtained through point-wise fine-tuning.Moreover,the operator is capable of analyzing the effect of dopant diffusion in manufacturing.展开更多
The warming and thawing of permafrost are the primary factors that impact the stability of embankments in cold regions.However,due to uncertainties in thermal boundaries and soil properties,the stochastic modeling of ...The warming and thawing of permafrost are the primary factors that impact the stability of embankments in cold regions.However,due to uncertainties in thermal boundaries and soil properties,the stochastic modeling of thermal regimes is challenging and computationally expensive.To address this,we propose a knowledge-integrated deep learning method for predicting the stochastic thermal regime of embankments in permafrost regions.Geotechnical knowledge is embedded in the training data through numerical modeling,while the neural network learns the mapping from the thermal boundary and soil property fields to the temperature field.The effectiveness of our method is verified in comparison with monitoring data and numerical analysis results.Experimental results show that the proposed method achieves good accuracy with small coefficient of variation.It still provides satisfactory accuracy as the coefficient of variation increases.The proposed knowledge-integrated deep learning method provides an efficient approach to predict the stochastic thermal regime of heterogeneous embankments.It can also be used in other permafrost engineering investigations that require stochastic numerical modeling.展开更多
Recent progress in topology optimization(TO)has seen a growing integration of machine learning to accelerate computation.Among these,online learning stands out as a promising strategy for large-scale TO tasks,as it el...Recent progress in topology optimization(TO)has seen a growing integration of machine learning to accelerate computation.Among these,online learning stands out as a promising strategy for large-scale TO tasks,as it eliminates the need for pre-collected training datasets by updating surrogate models dynamically using intermediate optimization data.Stress-constrained lightweight design is an important class of problem with broad engineering relevance.Most existing frameworks use pixel or voxel-based representations and employ the finite element method(FEM)for analysis.The limited continuity across finite elements often compromises the accuracy of stress evaluation.To overcome this limitation,isogeometric analysis is employed as it enables smooth representation of structures and thus more accurate stress computation.However,the complexity of the stress-constrained design problem together with the isogeometric representation results in a large computational cost.This work proposes a multi-grid,single-mesh online learning framework for isogeometric topology optimization(ITO),leveraging the Fourier Neural Operator(FNO)as a surrogate model.Operating entirely within the isogeometric analysis setting,the framework provides smooth geometry representation and precise stress computation,without requiring traditional mesh generation.A localized training approach is employed to enhance scalability,while a multi-grid decomposition scheme incorporates global structural context into local predictions to boost FNO accuracy.By learning the mapping from spatial features to sensitivity fields,the framework enables efficient single-resolution optimization,avoiding the computational burden of two-resolution simulations.The proposed method is validated through 2D stress-constrained design examples,and the effect of key parameters is studied.展开更多
Transformer neural operators have recently become an effective approach for surrogate modeling of systems governed by partial differential equations(PDEs).In this paper,we introduce a modified implicit factorized tran...Transformer neural operators have recently become an effective approach for surrogate modeling of systems governed by partial differential equations(PDEs).In this paper,we introduce a modified implicit factorized transformer(IFactFormer-m)model,replacing the original chained factorized attention with parallel factorized attention.The IFactFormer-m model successfully performs long-term predictions for turbulent channel flow.In contrast,the original IFactFormer(IFactFormer-o),Fourier neural operator(FNO),and implicit Fourier neural operator(IFNO)exhibit a poor performance.Turbulent channel flows are simulated by direct numerical simulation using fine grids at friction Reynolds numbers Re_τ≈180,395,590,and filtered to coarse grids for training neural operator.The neural operator takes the current flow field as input and predicts the flow field at the next time step,and long-term prediction is achieved in the posterior through an autoregressive approach.The results show that IFactFormer-m,compared with other neural operators and the traditional large eddy simulation(LES)methods,including the dynamic Smagorinsky model(DSM)and the wall-adapted local eddy-viscosity(WALE)model,reduces prediction errors in the short term,and achieves stable and accurate long-term prediction of various statistical properties and flow structures,including the energy spectrum,mean streamwise velocity,root mean square(RMS)values of fluctuating velocities,Reynolds shear stress,and spatial structures of instantaneous velocity.Moreover,the trained IFactFormer-m is much faster than traditional LES methods.By analyzing the attention kernels,we elucidate why IFactFormer-m converges faster and achieves a stable and accurate long-term prediction compared with IFactFormer-o.Code and data are available at:https://github.com/huiyu-2002/IFactFormer-m.展开更多
In this paper,we propose a machine learning approach via model-operatordata network(MOD-Net)for solving PDEs.A MOD-Net is driven by a model to solve PDEs based on operator representationwith regularization fromdata.Fo...In this paper,we propose a machine learning approach via model-operatordata network(MOD-Net)for solving PDEs.A MOD-Net is driven by a model to solve PDEs based on operator representationwith regularization fromdata.For linear PDEs,we use a DNN to parameterize the Green’s function and obtain the neural operator to approximate the solution according to the Green’s method.To train the DNN,the empirical risk consists of the mean squared loss with the least square formulation or the variational formulation of the governing equation and boundary conditions.For complicated problems,the empirical risk also includes a fewlabels,which are computed on coarse grid points with cheap computation cost and significantly improves the model accuracy.Intuitively,the labeled dataset works as a regularization in addition to the model constraints.The MOD-Net solves a family of PDEs rather than a specific one and is much more efficient than original neural operator because few expensive labels are required.We numerically show MOD-Net is very efficient in solving Poisson equation and one-dimensional radiative transfer equation.For nonlinear PDEs,the nonlinear MOD-Net can be similarly used as an ansatz for solving nonlinear PDEs,exemplified by solving several nonlinear PDE problems,such as the Burgers equation.展开更多
基金supported by the National Natural Science Foundation of China(Grant Nos.42005003 and 41475094)National Key R&D Program of China(Grant No.2018YFC1506704).
文摘With the growing demand for high-precision flow field simulations in computational science and engineering,the super-resolution reconstruction of physical fields has attracted considerable research interest.However,tradi-tional numerical methods often entail high computational costs,involve complex data processing,and struggle to capture fine-scale high-frequency details.To address these challenges,we propose an innovative super-resolution reconstruction framework that integrates a Fourier neural operator(FNO)with an enhanced diffusion model.The framework employs an adaptively weighted FNO to process low-resolution flow field inputs,effectively capturing global dependencies and high-frequency features.Furthermore,a residual-guided diffusion model is introduced to further improve reconstruction performance.This model uses a Markov chain for noise injection in phys-ical fields and integrates a reverse denoising procedure,efficiently solved by an adaptive time-step ordinary differential equation solver,thereby ensuring both stability and computational efficiency.Experimental results demonstrate that the proposed framework significantly outperforms existing methods in terms of accuracy and efficiency,offering a promising solution for fine-grained data reconstruction in scientific simulations.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.42005003 and 41475094)。
文摘Nonlinear science is a fundamental area of physics research that investigates complex dynamical systems which are often characterized by high sensitivity and nonlinear behaviors.Numerical simulations play a pivotal role in nonlinear science,serving as a critical tool for revealing the underlying principles governing these systems.In addition,they play a crucial role in accelerating progress across various fields,such as climate modeling,weather forecasting,and fluid dynamics.However,their high computational cost limits their application in high-precision or long-duration simulations.In this study,we propose a novel data-driven approach for simulating complex physical systems,particularly turbulent phenomena.Specifically,we develop an efficient surrogate model based on the wavelet neural operator(WNO).Experimental results demonstrate that the enhanced WNO model can accurately simulate small-scale turbulent flows while using lower computational costs.In simulations of complex physical fields,the improved WNO model outperforms established deep learning models,such as U-Net,Res Net,and the Fourier neural operator(FNO),in terms of accuracy.Notably,the improved WNO model exhibits exceptional generalization capabilities,maintaining stable performance across a wide range of initial conditions and high-resolution scenarios without retraining.This study highlights the significant potential of the enhanced WNO model for simulating complex physical systems,providing strong evidence to support the development of more efficient,scalable,and high-precision simulation techniques.
基金The National Natural Science Foundation of China under contract No.42425606the Basic Scientific Fund for the National Public Research Institute of China(Shu-Xingbei Young Talent Program)under contract No.2023S01+1 种基金the Ocean Decade International Cooperation Center Scientific and Technological Cooperation Project under contract No.GHKJ2024005China-Korea Joint Ocean Research Center Project under contract Nos PI-20240101(China)and 20220407(Korea).
文摘In oceanic and atmospheric science,finer resolutions have become a prevailing trend in all aspects of development.For high-resolution fluid flow simulations,the computational costs of widely used numerical models increase significantly with the resolution.Artificial intelligence methods have attracted increasing attention because of their high precision and fast computing speeds compared with traditional numerical model methods.The resolution-independent Fourier neural operator(FNO)presents a promising solution to the still challenging problem of high-resolution fluid flow simulations based on low-resolution data.Accordingly,we assess the potential of FNO for high-resolution fluid flow simulations using the vorticity equation as an example.We assess and compare the performance of FNO in multiple high-resolution tests varying the amounts of data and the evolution durations.When assessed with finer resolution data(even up to number of grid points with 1280×1280),the FNO model,trained at low resolution(number of grid points with 64×64)and with limited data,exhibits a stable overall error and good accuracy.Additionally,our work demonstrates that the FNO model takes less time than the traditional numerical method for high-resolution simulations.This suggests that FNO has the prospect of becoming a cost-effective and highly precise model for high-resolution simulations in the future.Moreover,FNO can make longer high-resolution predictions while training with less data by superimposing vorticity fields from previous time steps as input.A suitable initial learning rate can be set according to the frequency principle,and the time intervals of the dataset need to be adjusted according to the spatial resolution of the input when training the FNO model.Our findings can help optimize FNO for future fluid flow simulations.
基金supported by the National Natural Science Foundation of China(Nos.91952104,92052301,12172161,and 12161141017)National Numerical Windtunnel Project(No.NNW2019ZT1-A04)+4 种基金Shenzhen Science and Technology Program(No.KQTD20180411143441009)Key Special Project for Introduced Talents Team of Southern Marine Science and Engineering Guangdong Laboratory(Guangzhou)(No.GML2019ZD0103)CAAI-Huawei Mind Spore open Fundand by Department of Science and Technology of Guangdong Province(No.2019B21203001)supported by Center for Computational Science and Engineering of Southern University of Science and Technology。
文摘Fourier neural operator(FNO)model is developed for large eddy simulation(LES)of three-dimensional(3D)turbulence.Velocity fields of isotropic turbulence generated by direct numerical simulation(DNS)are used for training the FNO model to predict the filtered velocity field at a given time.The input of the FNO model is the filtered velocity fields at the previous several time-nodes with large time lag.In the a posteriori study of LES,the FNO model performs better than the dynamic Smagorinsky model(DSM)and the dynamic mixed model(DMM)in the prediction of the velocity spectrum,probability density functions(PDFs)of vorticity and velocity increments,and the instantaneous flow structures.Moreover,the proposed model can significantly reduce the computational cost,and can be well generalized to LES of turbulence at higher Taylor-Reynolds numbers.
基金the NSFC under Grant Nos.11925108 and 11731014the NSFC under Grant No.11975306
文摘In this paper,we develop the deep learning-based Fourier neural operator(FNO)approach to find parametric mappings,which are used to approximately display abundant wave structures in the nonlinear Schr?dinger(NLS)equation,Hirota equation,and NLS equation with the generalized PT-symmetric Scarf-II potentials.Specifically,we analyze the state transitions of different types of solitons(e.g.bright solitons,breathers,peakons,rogons,and periodic waves)appearing in these complex nonlinear wave equations.By checking the absolute errors between the predicted solutions and exact solutions,we can find that the FNO with the Ge Lu activation function can perform well in all cases even though these solution parameters have strong influences on the wave structures.Moreover,we find that the approximation errors via the physics-informed neural networks(PINNs)are similar in magnitude to those of the FNO.However,the FNO can learn the entire family of solutions under a given distribution every time,while the PINNs can only learn some specific solution each time.The results obtained in this paper will be useful for exploring physical mechanisms of soliton excitations in nonlinear wave equations and applying the FNO in other nonlinear wave equations.
文摘An efficient data-driven approach for predicting steady airfoil flows is proposed based on the Fourier neural operator(FNO),which is a new framework of neural networks.Theoretical reasons and experimental results are provided to support the necessity and effectiveness of the improvements made to the FNO,which involve using an additional branch neural operator to approximate the contribution of boundary conditions to steady solutions.The proposed approach runs several orders of magnitude faster than the traditional numerical methods.The predictions for flows around airfoils and ellipses demonstrate the superior accuracy and impressive speed of this novel approach.Furthermore,the property of zero-shot super-resolution enables the proposed approach to overcome the limitations of predicting airfoil flows with Cartesian grids,thereby improving the accuracy in the near-wall region.There is no doubt that the unprecedented speed and accuracy in forecasting steady airfoil flows have massive benefits for airfoil design and optimization.
基金supported by the National Natural Science Foundation of China(Grant Nos.U23A2069,12372288,12388101,and 92152301)the Jilin Province Science and Technology Development Program,China(Grant No.20220301013GX)。
文摘Traditional aerodynamic optimization coupled with computational fluid dynamics is associated with a high computational cost.Surrogate models based on deep learning methods can rapidly predict flow fields from the grid input but often suffer from poor accuracy and generalizability.This study introduces a modified Fourier neural operator for flow field prediction.Unlike most convolution-based models,the Fourier neural operator learns the solution operator directly in the function space,enhancing predictive accuracy and generalizability.The proposed model incorporates a shallow feature extractor,a boundary variable finetuner,and several physical priors,including the initial flow field and boundary conditions.The model is trained on uniformly parameterized algebraic grids to accelerate grid generation in aerodynamic optimization.The prediction error for the flow field and force coefficients on the validation and test sets is reduced by 70%to 90%compared with that of the previous convolutional model.The proposed model can make precise predictions for supercritical airfoils under typical working conditions,with a drag coefficient error of approximately 1 drag count on the validation set,and generalizes better than previous convolution-based methods do on extrapolative inflow conditions and airfoils.
基金supported by the National Research Foundation of Korea(NRF)(No.RS-2023-00222166 and No.RS-2023-00247245)the InnoCORE program(No.N10250154),both funded by the Korean government(MSIT).
文摘Accurate characterization of temperature-dependent thermoelectric properties(TEPs),such as thermal conductivity and the Seebeck coefficient,is essential for modeling and design of thermoelectric devices.However,nonlinear temperature dependence and coupled transport behavior make forward simulation and inverse identification challenging under sparse measurements.We present a physics-informed machine learning framework combining physics-informed neural networks(PINN)and neural operators(PINO)for solving forward and inverse problems in thermoelectric systems.PINN enables field reconstruction and property inference by embedding governing equations into the loss function,while PINO generalizes across materials without retraining.Trained on simulated data for 20 p-type materials and tested on 60 unseen materials,PINO accurately infers TEPs using only sparse temperature and voltage data.This framework provides a scalable,dataefficient,and generalizable solution for thermoelectric property identification,facilitating highthroughput screening and inverse design of advanced thermoelectric materials.
文摘Traditionally,classical numerical schemes have been employed to solve partial differential equations(PDEs)using computational methods.Recently,neural network-based methods have emerged.Despite these advancements,neural networkbased methods,such as physics-informed neural networks(PINNs)and neural operators,exhibit deficiencies in robustness and generalization.To address these issues,numerous studies have integrated classical numerical frameworks with machine learning techniques,incorporating neural networks into parts of traditional numerical methods.In this study,we focus on hyperbolic conservation laws by replacing traditional numerical fluxes with neural operators.To this end,we developed loss functions inspired by established numerical schemes related to conservation laws and approximated numerical fluxes using Fourier neural operators(FNOs).Our experiments demonstrated that our approach combines the strengths of both traditional numerical schemes and FNOs,outperforming standard FNO methods in several respects.For instance,we demonstrate that our method is robust,has resolution invariance,and is feasible as a data-driven method.In particular,our method can make continuous predictions over time and exhibits superior generalization capabilities with out-of-distribution(OOD)samples,which are challenges that existing neural operator methods encounter.
基金funding from the project“SPEED”(03XP0585)funded by the German Federal Ministry of ResearchTech-nology and Space(BMFTR)and the project“ADMirABLE”(03ETE053E)funded by the German Federal Ministry for Economic Affairs and Energy(BMWE)support of Shell Research UK Ltd.for the Ph.D.studentship of Amir Ali Panahi and the EPSRC Faraday Institution Multi-Scale Modelling Project(FIRG084).
文摘Digital twins of lithium-ion batteries are increasingly used to enable predictive monitoring,control,and design at system scale.Increasing their capabilities involves improving their physical fidelity while maintaining sub-millisecond computational speed.In this work,we introduce machine learning surrogates that learn physical dynamics.Specifically,we benchmark three operator-learning surrogates for the Single Particle Model(SPM):Deep Operator Networks(DeepONets),Fourier Neural Operators(FNOs)and a newly proposed parameter-embedded Fourier Neural Operator(PE-FNO),which conditions each spectral layer on particle radius and solid-phase diffusivity.We extend the comparison to classical machine-learning baselines by including U-Nets.Models are trained on simulated trajectories spanning four current families(constant,triangular,pulse-train,and Gaussian-random-field)and a full range of State-of-Charge(SOC)(0%to 100%).DeepONet accurately replicates constant-current behaviour but struggles with more dynamic loads.The basic FNO maintains mesh invariance and keeps concentration errors below 1%,with voltage mean-absolute errors under 1.7mV across all load types.Introducing parameter embedding marginally increases error but enables generalisation to varying radii and diffusivities.PE-FNO executes approximately 200 times faster than a 16-thread SPM solver.Consequently,PE-FNO’s capabilities in inverse tasks are explored in a parameter estimation task with Bayesian optimisation,recovering anode and cathode diffusivities with 1.14%and 8.4%mean absolute percentage error,respectively,and 0.5918 percentage points higher error in comparison with classical methods.These results pave the way for neural operators to meet the accuracy,speed and parametric flexibility demands of real-time battery management,design-of-experiments and large-scale inference.PE-FNO outperforms conventional neural surrogates,offering a practical path towards high-speed and high-fidelity electrochemical digital twins.
基金supported by the National Natural Science Foundation Major Program of China(No.12494544)the National Natural Science Foundation General Program of China(No.12171039)+1 种基金the New Cornerstone Science Foundation through the XPLORER PRIZE and Sino-German Center Mobility Programme(No.M-0548)the Shanghai Science and Technology Program(No.21JC1400600)。
文摘The neural operator theory ofers a promising framework for efficiently solving complex systems governed by partial diferential equations(PDEs for short).However,existing neural operators still face significant challenges when applied to spatiotemporal systems that evolve over large time scales,particularly those described by evolution PDEs with time-derivative terms.This paper introduces a novel neural operator designed explicitly for solving evolution equations based on the theory of operator semigroups.The proposed approach is an iterative algorithm where each computational unit,termed the single-step neural operator solver(SSNOS for short),approximates the solution operator for the initial-boundary value problem of semilinear evolution equations over a single time step.The SSNOS consists of both linear and nonlinear components:The linear part approximates the linear operator in the solution map;in contrast,the nonlinear part captures deviations in the solution function caused by the equations nonlinearities.To evaluate the performance of the algorithm,the authors conducted numerical experiments by solving the initial-boundary value problem for a two-dimensional semilinear hyperbolic equation.The experimental results demonstrate that their neural operator can efficiently and accurately approximate the true solution operator.Moreover,the model can achieve a relatively high approximation accuracy with simple pre-training.
基金supported by the National Natural Science Foundation of China[grant number 52078009]the Joint Research Project of the Wind Engineering Research Center,Tokyo Polytechnic University(MEXT(Japan)Promotion of the Distinctive Joint Research Center Program)[Joint Research Assignment number JURC 24242008].
文摘Indoor airflow distribution significantly influences temperature regulation,humidity control,and pollutant dispersion,directly impacting thermal comfort and occupant health.Accurate and efficient prediction of airflow fields is essential for optimizing ventilation systems and enabling real-time control.However,existing computational approaches for dynamic ventilation are computationally intensive and have limited generalization capabilities.This study leverages the Fourier neural operator(FNO),a method rooted in operator learning and Fourier transform principles,to develop a three-dimensional(3D)airflow simulation model capable of predicting velocity and its components.The model was trained using 200 s of sinusoidal ventilation data(amplitude:0.4)and evaluated under diverse air supply patterns,including sinusoidal(amplitude:0.8),intermittent,and stepwise periodic ventilation.Additionally,the model’s performance was assessed with low-resolution training data and further tested for recursive prediction accuracy.Results reveal that the FNO method achieves high accuracy,with a mean square error of 9.906×10^(-5)for sinusoidal amplitude 0.8 and 4.004×10^(-5)over 400 time steps for sinusoidal,intermittent,and stepwise conditions.Further evaluations,including tests on low-resolution training data and recursive prediction,were conducted to examine the model’s resolution invariance and assess its performance in iterative forecasting.These findings demonstrate the FNO method’s potential for robust,efficient prediction of 3D unsteady airflow fields,providing a pathway for real-time ventilation system optimization.
基金supported by the U.S.Air Force under agreement number FA865019-2-2204.
文摘Partial differential equations(PDEs)play a dominant role in themathematicalmodeling ofmany complex dynamical processes.Solving these PDEs often requires prohibitively high computational costs,especially when multiple evaluations must be made for different parameters or conditions.After training,neural operators can provide PDEs solutions significantly faster than traditional PDE solvers.In this work,invariance properties and computational complexity of two neural operators are examined for transport PDE of a scalar quantity.Neural operator based on graph kernel network(GKN)operates on graph-structured data to incorporate nonlocal dependencies.Here we propose a modified formulation of GKN to achieve frame invariance.Vector cloud neural network(VCNN)is an alternate neural operator with embedded frame invariance which operates on point cloud data.GKN-based neural operator demonstrates slightly better predictive performance compared to VCNN.However,GKN requires an excessively high computational cost that increases quadratically with the increasing number of discretized objects as compared to a linear increase for VCNN.
基金supported by the National Science Fund for Distinguished Young Scholars (51925505)the General Program of National Natural Science Foundation of China (52275491)+3 种基金the Major Program of the National Natural Science Foundation of China (52090052)the Joint Funds of the National Natural Science Foundation of China (U21B2081)the National Key R&D Program of China (2022YFB3402600)the New Cornerstone Science Foundation through the XPLORER PRIZE
文摘Learning mappings between functions(operators)defined on complex computational domains is a common theoretical challenge in machine learning.Existing operator learning methods mainly focus on regular computational domains,and have many components that rely on Euclidean structural data.However,many real-life operator learning problems involve complex computational domains such as surfaces and solids,which are non-Euclidean and widely referred to as Riemannian manifolds.Here,we report a new concept,neural operator on Riemannian manifolds(NORM),which generalises neural operator from Euclidean spaces to Riemannian manifolds,and can learn the operators defined on complex geometries while preserving the discretisation-independent model structure.NORM shifts the function-to-function mapping to finite-dimensional mapping in the Laplacian eigenfunctions’subspace of geometry,and holds universal approximation property even with only one fundamental block.The theoretical and experimental analyses prove the significant performance of NORM in operator learning and show its potential for many scientific discoveries and engineering applications.
基金Supported by the National Natural Science Foundation of China(61179041,61101240)the Zhejiang Provincial Natural Science Foundation of China(Y6110117)
文摘In this paper,the technique of approximate partition of unity is used to construct a class of neural networks operators with sigmoidal functions.Using the modulus of continuity of function as a metric,the errors of the operators approximating continuous functions defined on a compact interval are estimated.Furthmore,Bochner-Riesz means operators of double Fourier series are used to construct networks operators for approximating bivariate functions,and the errors of approximation by the operators are estimated.
基金supported by the National Natural Science Foundation of China(Grant Nos.U24B20133 and 62522104)the Beijing Nova Program(Grant No.20230484331).
文摘An efficient neural mode-solving operator is proposed for evaluating the propagation properties of optical fibers.By incorporating the governing Helmholtz equation into training,the working mechanism of the proposed operator adheres to the physics essence of fiber analysis.The training of the mode-solving operator adopts a hybrid physics-informed and data-driven approach,providing the advantages of strong physical consistency,enhanced prediction accuracy,and reduced data dependency in comparison with purely datadriven methods.Benefiting from the improvements in network input-output mapping formulation,the proposed operator offers broader applicability to different fiber types and greater flexibility for property optimization.Combined with the particle swarm optimization and refractive index optimization,the operator demonstrates its capacity for the inverse design of multi-step-index fibers(MSIFs)and graded-index fibers(GRIFs).For MSIFs,to ensure a low mode crosstalk for short-distance transmission systems,optimized refractive index profiles(RIPs)of both three-ring and four-ring structures are obtained from large structure parameter search spaces.For GRIFs,to ensure a low receiving complexity for long-haul transmission systems,optimized RIP with low root mean square mode group delay is obtained through point-wise fine-tuning.Moreover,the operator is capable of analyzing the effect of dopant diffusion in manufacturing.
基金funding support from the National Natural Science Foundation of China(Grant Nos.42277161,42230709).
文摘The warming and thawing of permafrost are the primary factors that impact the stability of embankments in cold regions.However,due to uncertainties in thermal boundaries and soil properties,the stochastic modeling of thermal regimes is challenging and computationally expensive.To address this,we propose a knowledge-integrated deep learning method for predicting the stochastic thermal regime of embankments in permafrost regions.Geotechnical knowledge is embedded in the training data through numerical modeling,while the neural network learns the mapping from the thermal boundary and soil property fields to the temperature field.The effectiveness of our method is verified in comparison with monitoring data and numerical analysis results.Experimental results show that the proposed method achieves good accuracy with small coefficient of variation.It still provides satisfactory accuracy as the coefficient of variation increases.The proposed knowledge-integrated deep learning method provides an efficient approach to predict the stochastic thermal regime of heterogeneous embankments.It can also be used in other permafrost engineering investigations that require stochastic numerical modeling.
基金supported by the Hong Kong Research Grants under Competitive Earmarked Research Grant No.16206320.
文摘Recent progress in topology optimization(TO)has seen a growing integration of machine learning to accelerate computation.Among these,online learning stands out as a promising strategy for large-scale TO tasks,as it eliminates the need for pre-collected training datasets by updating surrogate models dynamically using intermediate optimization data.Stress-constrained lightweight design is an important class of problem with broad engineering relevance.Most existing frameworks use pixel or voxel-based representations and employ the finite element method(FEM)for analysis.The limited continuity across finite elements often compromises the accuracy of stress evaluation.To overcome this limitation,isogeometric analysis is employed as it enables smooth representation of structures and thus more accurate stress computation.However,the complexity of the stress-constrained design problem together with the isogeometric representation results in a large computational cost.This work proposes a multi-grid,single-mesh online learning framework for isogeometric topology optimization(ITO),leveraging the Fourier Neural Operator(FNO)as a surrogate model.Operating entirely within the isogeometric analysis setting,the framework provides smooth geometry representation and precise stress computation,without requiring traditional mesh generation.A localized training approach is employed to enhance scalability,while a multi-grid decomposition scheme incorporates global structural context into local predictions to boost FNO accuracy.By learning the mapping from spatial features to sensitivity fields,the framework enables efficient single-resolution optimization,avoiding the computational burden of two-resolution simulations.The proposed method is validated through 2D stress-constrained design examples,and the effect of key parameters is studied.
基金supported by the National Natural Science Foundation of China(Grant Nos.12172161,12302283,92052301,and 12161141017)the NSFC Basic Science Center Program(Grant No.11988102)+3 种基金the Shenzhen Science and Technology Program(Grant No.KQTD20180411143441009)the Department of Science and Technology of Guangdong Province(Grant Nos.2019B21203001,2020B1212030001,and 2023B1212060001)supported by Center for Computational Science and Engineering of Southern University of Science and Technologyby National Center for Applied Mathematics Shenzhen(NCAMS)。
文摘Transformer neural operators have recently become an effective approach for surrogate modeling of systems governed by partial differential equations(PDEs).In this paper,we introduce a modified implicit factorized transformer(IFactFormer-m)model,replacing the original chained factorized attention with parallel factorized attention.The IFactFormer-m model successfully performs long-term predictions for turbulent channel flow.In contrast,the original IFactFormer(IFactFormer-o),Fourier neural operator(FNO),and implicit Fourier neural operator(IFNO)exhibit a poor performance.Turbulent channel flows are simulated by direct numerical simulation using fine grids at friction Reynolds numbers Re_τ≈180,395,590,and filtered to coarse grids for training neural operator.The neural operator takes the current flow field as input and predicts the flow field at the next time step,and long-term prediction is achieved in the posterior through an autoregressive approach.The results show that IFactFormer-m,compared with other neural operators and the traditional large eddy simulation(LES)methods,including the dynamic Smagorinsky model(DSM)and the wall-adapted local eddy-viscosity(WALE)model,reduces prediction errors in the short term,and achieves stable and accurate long-term prediction of various statistical properties and flow structures,including the energy spectrum,mean streamwise velocity,root mean square(RMS)values of fluctuating velocities,Reynolds shear stress,and spatial structures of instantaneous velocity.Moreover,the trained IFactFormer-m is much faster than traditional LES methods.By analyzing the attention kernels,we elucidate why IFactFormer-m converges faster and achieves a stable and accurate long-term prediction compared with IFactFormer-o.Code and data are available at:https://github.com/huiyu-2002/IFactFormer-m.
基金sponsored by the National Key R&D Program of China Grant No.2019YFA0709503(Z.X.)and No.2020YFA0712000(Z.M.)the Shanghai Sailing Program(Z.X.)+9 种基金the Natural Science Foundation of Shanghai Grant No.20ZR1429000(Z.X.)the National Natural Science Foundation of China Grant No.62002221(Z.X.)the National Natural Science Foundation of China Grant No.12101401(T.L.)the National Natural Science Foundation of China Grant No.12101402(Y.Z.)Shanghai Municipal of Science and Technology Project Grant No.20JC1419500(Y.Z.)the Lingang Laboratory Grant No.LG-QS-202202-08(Y.Z.)the National Natural Science Foundation of China Grant No.12031013(Z.M.)Shanghai Municipal of Science and Technology Major Project No.2021SHZDZX0102the HPC of School of Mathematical Sciencesthe Student Innovation Center at Shanghai Jiao Tong University.
文摘In this paper,we propose a machine learning approach via model-operatordata network(MOD-Net)for solving PDEs.A MOD-Net is driven by a model to solve PDEs based on operator representationwith regularization fromdata.For linear PDEs,we use a DNN to parameterize the Green’s function and obtain the neural operator to approximate the solution according to the Green’s method.To train the DNN,the empirical risk consists of the mean squared loss with the least square formulation or the variational formulation of the governing equation and boundary conditions.For complicated problems,the empirical risk also includes a fewlabels,which are computed on coarse grid points with cheap computation cost and significantly improves the model accuracy.Intuitively,the labeled dataset works as a regularization in addition to the model constraints.The MOD-Net solves a family of PDEs rather than a specific one and is much more efficient than original neural operator because few expensive labels are required.We numerically show MOD-Net is very efficient in solving Poisson equation and one-dimensional radiative transfer equation.For nonlinear PDEs,the nonlinear MOD-Net can be similarly used as an ansatz for solving nonlinear PDEs,exemplified by solving several nonlinear PDE problems,such as the Burgers equation.
