# A finite element-based physics-informed operator learning framework for spatiotemporal partial differential equations on arbitrary domains

arXiv: https://arxiv.org/abs/2405.12465 · HTML: https://mlsanity.com/paper/2405.12465/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Builds on 41 of its 82 references that are in the map.
- First posted on arXiv 21 May 2024.

## Builds on (most cited first, 30 of 41)

- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domain](https://mlsanity.com/paper/2004.13145/) (2020)
- [Modeling the Dynamics of PDE Systems with Physics-Constrained Deep Auto-Regressive Networks](https://mlsanity.com/paper/1906.05747/) (2019)
- [An Energy Approach to the Solution of Partial Differential Equations in Computational Mechanics via Machine Learning: Concepts, Implementation and Applications](https://mlsanity.com/paper/1908.10407/) (2019)
- [Challenges in Training PINNs: A Loss Landscape Perspective](https://mlsanity.com/paper/2402.01868/) (2024)
- [PhyCRNet: Physics-informed Convolutional-Recurrent Network for Solving Spatiotemporal PDEs](https://mlsanity.com/paper/2106.14103/) (2021)
- [A physics-informed variational DeepONet for predicting the crack path in brittle materials](https://mlsanity.com/paper/2108.06905/) (2021)
- [Improved architectures and training algorithms for deep operator networks](https://mlsanity.com/paper/2110.01654/) (2021)
- [A Mathematical Guide to Operator Learning](https://mlsanity.com/paper/2312.14688/) (2023)
- [Physics-informed neural networks for solving Reynolds-averaged Navierunicodex2013Stokes equations](https://mlsanity.com/paper/2107.10711/) (2021)
- [DeepGreen: Deep Learning of Green's Functions for Nonlinear Boundary Value Problems](https://mlsanity.com/paper/2101.07206/) (2020)
- [Physics-informed deep learning for incompressible laminar flows](https://mlsanity.com/paper/2002.10558/) (2020)
- [A Physics-Informed Machine Learning Approach for Solving Heat Transfer Equation in Advanced Manufacturing and Engineering Applications](https://mlsanity.com/paper/2010.02011/) (2020)
- [Transfer learning based physics-informed neural networks for solving inverse problems in engineering structures under different loading scenarios](https://mlsanity.com/paper/2205.07731/) (2022)
- [Data-driven discovery of Green's functions with human-understandable deep learning](https://mlsanity.com/paper/2105.00266/) (2021)
- [Multi-resolution partial differential equations preserved learning framework for spatiotemporal dynamics](https://mlsanity.com/paper/2205.03990/) (2022)
- [Simulating progressive intramural damage leading to aortic dissection using an operator-regression neural network](https://mlsanity.com/paper/2108.11985/) (2021)
- [A mixed formulation for physics-informed neural networks as a potential solver for engineering problems in heterogeneous domains: comparison with finite element method](https://mlsanity.com/paper/2206.13103/) (2022)
- [Novel DeepONet architecture to predict stresses in elastoplastic structures with variable complex geometries and loads](https://mlsanity.com/paper/2306.03645/) (2023)
- [Physics-informed Convolutional Neural Networks for Temperature Field Prediction of Heat Source Layout without Labeled Data](https://mlsanity.com/paper/2109.12482/) (2021)
- [Meshless physics-informed deep learning method for three-dimensional solid mechanics](https://mlsanity.com/paper/2012.01547/) (2020)
- [Phase-Field DeepONet: Physics-informed deep operator neural network for fast simulations of pattern formation governed by gradient flows of free-energy functionals](https://mlsanity.com/paper/2302.13368/) (2023)
- [Hybrid FEM-NN models: Combining artificial neural networks with the finite element method](https://mlsanity.com/paper/2101.00962/) (2021)
- [Stress field prediction in fiber-reinforced composite materials using a deep learning approach](https://mlsanity.com/paper/2111.05271/) (2021)
- [Learning in latent spaces improves the predictive accuracy of deep neural operators](https://mlsanity.com/paper/2304.07599/) (2023)
- [Lossless Multi-Scale Constitutive Elastic Relations with Artificial Intelligence](https://mlsanity.com/paper/2108.02837/) (2021)

## Built on by (most cited first, 15 of 15)

- [Neural-Initialized Newton: Accelerating Nonlinear Finite Elements via Operator Learning](https://mlsanity.com/paper/2511.06802/) (2025)
- [A physics-informed transformer neural operator for learning generalized solutions of initial boundary value problems](https://mlsanity.com/paper/2412.09009/) (2024)
- [A Physics-Informed Meta-Learning Framework for the Continuous Solution of Parametric PDEs on Arbitrary Geometries](https://mlsanity.com/paper/2504.02459/) (2025)
- [Scale-PINN: Learning Efficient Physics-Informed Neural Networks Through Sequential Correction](https://mlsanity.com/paper/2602.19475/) (2026)
- [Projection Methods for Operator Learning and Universal Approximation](https://mlsanity.com/paper/2406.12264/) (2024)
- [Preconditioned FEM-based Neural Networks for Solving Incompressible Fluid Flows and Related Inverse Problems](https://mlsanity.com/paper/2409.04067/) (2024)
- [A discrete physics-informed training for projection-based reduced order models with neural networks](https://mlsanity.com/paper/2504.13875/) (2025)
- [A meshfree exterior calculus for generalizable and data-efficient learning of physics from point clouds](https://mlsanity.com/paper/2605.08436/) (2026)
- [Preconditioned Physics-Informed Neural Operator Training](https://mlsanity.com/paper/2609.36216/) (2026)
- [Prescribed-Basis Coefficient-to-Coefficient Neural Operator for Partial Differential Equations](https://mlsanity.com/paper/2510.10350/) (2025)
- [High-Fidelity Modeling of Stochastic Chemical Dynamics on Complex Manifolds: A Multi-Scale SIREN-PINN Framework for the Curvature-Perturbed Ginzburg-Landau Equation](https://mlsanity.com/paper/2601.08104/) (2026)
- [Tackling multiphysics problems via finite element-guided physics-informed operator learning](https://mlsanity.com/paper/2603.01420/) (2026)
- [Physics-informed post-processing of stabilized finite element solutions for transient convection-dominated problems](https://mlsanity.com/paper/2603.03259/) (2026)
- [DeepRitzSplit Neural Operator for Phase-Field Models via Energy Splitting](https://mlsanity.com/paper/2604.18261/) (2026)
- [Learning the Constitutive Behavior of Materials via Neural Operators and Causal Attention: Case Studies in Plasticity and Damage](https://mlsanity.com/paper/2609.02194/) (2026)

## Abstract (arXiv)

We propose a novel finite element-based physics-informed operator learning framework that allows for predicting spatiotemporal dynamics governed by partial differential equations (PDEs). The proposed framework employs a loss function inspired by the finite element method (FEM) with the implicit Euler time integration scheme. A transient thermal conduction problem is considered to benchmark the performance. The proposed operator learning framework takes a temperature field at the current time step as input and predicts a temperature field at the next time step. The Galerkin discretized weak formulation of the heat equation is employed to incorporate physics into the loss function, which is coined finite operator learning (FOL). Upon training, the networks successfully predict the temperature evolution over time for any initial temperature field at high accuracy compared to the FEM solution. The framework is also confirmed to be applicable to a heterogeneous thermal conductivity and arbitrary geometry. The advantages of FOL can be summarized as follows: First, the training is performed in an unsupervised manner, avoiding the need for a large data set prepared from costly simulations or experiments. Instead, random temperature patterns generated by the Gaussian random process and the Fourier series, combined with constant temperature fields, are used as training data to cover possible temperature cases. Second, shape functions and backward difference approximation are exploited for the domain discretization, resulting in a purely algebraic equation. This enhances training efficiency, as one avoids time-consuming automatic differentiation when optimizing weights and biases while accepting possible discretization errors. Finally, thanks to the interpolation power of FEM, any arbitrary geometry can be handled with FOL, which is crucial to addressing various engineering application scenarios.