Tackling multiphysics problems via finite element-guided physics-informed operator learning
Authors: Yusuke Yamazaki, Reza Najian Asl, Markus Apel, Mayu Muramatsu, Shahed Rezaei
Organizations: Graduate School of Science and Technology, Keio University, Hiyoshi3-14-1,Kohoku-ku, Yokohama 223-8522, Japan · Chair of Structural Analysis, Technical University of Munich, Arcisstraße 21, 80333 Munich, Germany · ACCESS e.V., Intzestr. 5, 52072 Aachen, Germany · Department of Mechanical Engineering, Keio University, Hiyoshi3-14-1,Kohoku-ku, Yokohama 223-8522, Japan
This work presents a finite element-guided physics-informed operator learning framework for multiphysics problems with coupled partial differential equations (PDEs) on arbitrary domains. The proposed framework learns an operator from the input space to the solution space with a weighted residual formulation based on the finite element method, enabling discretization-independent prediction beyond the training resolution without relying on labeled simulation data. The present framework for multiphysics problems is implemented in Folax, a JAX-based operator learning platform, and is verified on nonlinear coupled thermo-mechanical problems. Two- and three-dimensional representative volume elements with varying heterogeneous microstructures, and a close-to-reality industrial casting example under varying boundary conditions are investigated as the example problems. We investigate the potential of several neural operators combined with the proposed finite element-guided approach, including Fourier neural operators (FNOs), deep operator networks (DeepONets), and a newly proposed implicit finite operator learning (iFOL) approach based on conditional neural fields. The results demonstrate that FNOs yield highly accurate solution operators on regular domains, where the global features can be efficiently learned in the spectral domain, and iFOL offers efficient parametric operator learning capabilities for complex and irregular geometries. Furthermore, studies on training strategies, network decomposition, and training sample quality reveal that a monolithic training strategy using a single network is sufficient for accurate predictions, while training sample quality strongly influences performance. Overall, the present approach highlights the potential of physics-informed operator learning with a finite element-based loss as a unified and scalable approach for coupled multiphysics simulations.
Figures & tables
Figure 1: Overall idea of the present research: addressing multiphysics problems across scales with a novel and scalable physics-informed operator learning framework, where finite element residuals are directly used for parametric training of deep learning models. The final goal would be to enable strong coupling of multiphysics problems between macro and microscale as illustrated on the bottom. As a first step, the present work demonstrates the performance of the proposed framework on typical multiphysics problems on both scales.
Figure 2: Temperature-dependent material properties utilized in this study.
Figure 3: Schematic of the three operator learning backbones employed in this study: Fourier Neural Operator (FNO), Deep Operator Network (DeepONet), and implicit Finite Operator Learning (iFOL).
Figure 4: Domain and boundary conditions of the three examples. (a) Two-dimensional squared-domain problem, (b) three-dimensional RVE problem with varying material properties, and (c) casting example problem with varying boundary conditions.
Figure 5: Examples of training (left) and test samples (right) employed for the two-dimensional square-domain problem.
Figure 6
Figure 8: Comparison of the prediction and reference solution from NFEM in the two-dimensional square domain problem for a representative case of the in-distribution and gyroid test samples.
Figure 9: Comparison of the prediction and reference solution from NFEM in the two-dimensional square domain problem for a representative case of the polycrystal and dualphase test samples.
Figure 10: Comparison of the three different network decomposition strategies for FNO-based multiphysics operator learning.
Figure 11: Relative L2 error statistics over 50 samples on four different test cases with three different network decomposition strategies.
Figure 12: Relative L2 error statistics over 50 samples on four different test cases with the monolithic and staggered training schemes.
Figure 13: Relative L2 error statistics over 50 samples on four different test cases on two different training samples with different frequency sets in the generation of training samples based on the Fourier series.
Figure 14: Relative L2 error statistics over 50 samples on four different test cases on two different training samples with different numbers of training samples.
Figure 15: Comparison of average inference cost and calculation cost by NFEM over 50 cases on three different resolutions.
Location
x=0
x=Lx
y=0
y=Ly
z=0
z=Lz
T
0.5
0.0
free
free
free
free
Ux
0.0
0.1
free
free
free
free
Uy
0.0
0.1
free
free
free
free
Uz
0.0
0.1
free
free
free
free
Table 1: List of temperature and displacement boundary conditions for the three-dimensional nonlinear themo-mechanics on a RVE ; Here, T , Ux , Uy , Uz denote the prescribed temperature or displacements as Dirichlet boundary conditions in the x -, y -, and z -directions respectively, and "free" indicates Dirichlet boundary conditions are not enforced; Instead, natural boundary conditions are applied.
Figure 16
Figure 18: Comparison of the prediction and reference solution from NFEM in the three-dimensional RVE problem for a representative case of the in-distribution test cases.
Figure 19: Comparison of the prediction and reference solution from NFEM in the three-dimensional RVE problem for a representative case of the dual-phase structure test cases.
Location
Bottom of F-shaped part z=0.0
Surroundings of F-shaped part
Bottom of inlet part
Top of inlet part z=0.85
T
0.001
0.001
free
parametric
Ux
0.0
free
0.0
0.0
Uy
0.0
free
0.0
0.0
Uz
0.0
free
free
0.0
Table 2: List of boundary conditions for the three-dimensional nonlinear themo-mechanics on a casting geometry ; Here, T , Ux , Uy , Uz denote the prescribed temperature or displacements as Dirichlet boundary conditions in the x -, y -, and z -directions respectively, and "free" indicates natural boundary conditions are applied.
Figure 20: Relative L2 error statistics over 10 uniformly sampled test temperature values on two different operator learning architectures (DeepONet and iFOL) for the irregular casting example geometry.
Figure 21: Cross-section comparison of the prediction with iFOL and DeepONet-FOL against the reference NFEM solution at z=0.45 in the three-dimensional casting example problem for an unseen temperature case.
Figure 22: Comparison of the prediction with iFOL and reference solution from NFEM in the three-dimensional casting example problem for an unseen temperature case.
Figure 23: Comparison of the prediction with DeepONet-FOL and reference solution from NFEM in the three-dimensional casting example problem for an unseen temperature case.
Appendix figures & tables4 assets
Supplementary material from the paper’s appendix.
Appendix
Training parameter
FNO (Sec. 4.1 )
FNO (Sec. 4.2 )
Number of epochs
1000
1000
Optimizer
Adam
Adam
Grid in training
42×42
22×22×22
Grid in evaluation
84×84
44×44×44
Number of modes
(12,12)
(10,10,10)
Number of channels
24
8
Appendix
Table 1: Network hyperparameters for the 2D and 3D regular domain problems with Fourier neural operators.
Training parameter
DeepONet (Sec. 4.3 )
iFOL (Sec. 4.3 )
Number of epochs
10000
10000
Optimizer
Adam
Adam
Number of nodes
16322
16322
Branch net/Modulator for each layer
[512,512]
[128,128,128,128]
Trunk net/Synthesizer
[128,128,128,128]
[128,128,128,128,128,128]
Activation function
ReLU
Leaky ReLU
Appendix
Table 2: Network hyperparameters of DeepONet and iFOL for the three-dimensional casting example problem.
Figure 1: Relative L2 error statistics over 50 samples on four different test cases on different numbers of Fourier modes in FNO.
Figure 2: Relative L2 error statistics over 50 samples on four different test cases on different numbers of FNO layers.
Neural operators are typically trained in a supervised fashion, which requires a dataset to be generated with a classical solver. Training them physics-informed, i.e., purely from the governing equations, removes this large offline cost and allows fresh samples to be drawn at every optimization step, but has so far been limited to simplified problems and trails supervised training in accuracy. The obstacle is the ill-conditioning of physics-informed losses, which differential operators induce and which worsens as the discretization is refined. We therefore propose a preconditioned residual loss function and show mesh-independent conditioning for elliptic problems and greatly improved conditioning for saddle point problems. Realized through geometric and algebraic multigrid, the construction applies to linear and nonlinear equations, steady or time-dependent, on structured and unstructured meshes, is agnostic to the neural operator architecture, and adds no cost at inference. On the Poisson, Allen-Cahn and stationary Stokes equations, the resulting label-free training matches supervised training and is four to twenty-five times more accurate than previous physics-informed operator learning methods.
Recent advances in scientific machine learning (SciML) have enabled neural operators (NOs) to serve as powerful surrogates for modeling the dynamic evolution of physical systems governed by partial differential equations (PDEs). While existing approaches focus primarily on learning simulations from the target PDE, they often overlook more fundamental physical principles underlying these equations. Inspired by how numerical solvers are compatible with simulations of different settings of PDEs, we propose a multiphysics training framework that jointly learns from both the original PDEs and their simplified basic forms. Our framework enhances data efficiency, reduces predictive errors, and improves out-of-distribution (OOD) generalization, particularly in scenarios involving shifts of physical parameters and synthetic-to-real transfer. Our method is architecture-agnostic and demonstrates consistent improvements in normalized root mean square error (nRMSE) across a wide range of 1D/2D/3D PDE problems. Through extensive experiments, we show that explicit incorporation of fundamental physics knowledge significantly strengthens the generalization ability of neural operators. We will release models and codes at https://sites.google.com/view/sciml-fundemental-pde.
Siying Ma, Mehrdad M. Zadeh, Mauricio Soroco +3
Simon Fraser University · Georgia Institute of Technology
Neural operators have emerged as promising surrogate models for solving partial differential equations (PDEs), but struggle to generalise beyond training distributions and are often constrained to a fixed temporal discretisation. This work introduces a physics-informed training framework that addresses these limitations by decomposing PDEs using operator splitting methods, training separate neural operators to learn individual non-linear physical operators while approximating linear operators with fixed finite-difference convolutions. This modular mixture-of-experts architecture enables generalisation to novel physical regimes by explicitly encoding the underlying operator structure. We formulate the modelling task as a neural ordinary differential equation (ODE) where these learned operators constitute the right-hand side, enabling continuous-in-time predictions through standard ODE solvers and implicitly enforcing PDE constraints. Demonstrated on incompressible and compressible Navier--Stokes equations, our approach achieves better convergence and superior performance when generalising to unseen physics. The method remains parameter-efficient, enabling temporal extrapolation beyond training horizons, and provides interpretable components whose behaviour can be verified against known physics.
Vignesh Gopakumar, Ander Gray, Dan Giles +5
UCL Centre for Artificial Intelligence · UK Atomic Energy Authority · LIX, CNRS, ´Ecole Polytechnique +2