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Learning a Neural Solver for Parametric PDE to Enhance Physics-Informed Methods

Lise Le Boudec, Emmanuel de Bezenac, Louis Serrano, Ramon Daniel Regueiro-Espino, Yuan Yin, Patrick Gallinari

TL;DR

This work tackles the optimization bottlenecks of physics-informed PDE solvers by learning a neural optimizer that conditions gradient updates on PDE parameters, enabling rapid convergence for parametric PDEs. By combining a linear basis expansion for the solution with Fourier-based neural conditioning, the approach learns to transform ill-conditioned PDE residual landscapes into well-behaved ones, reducing required iterations from thousands to a handful at test time. Theoretical analysis links the learned solver to preconditioning, showing potential to achieve near-optimal convergence rates under linear assumptions, while extensive experiments demonstrate superior accuracy and fast inference across multiple PDE families. Overall, the method offers a practical path to efficient, parametric PDE solving that complements and often outperforms traditional PINN-based approaches.

Abstract

Physics-informed deep learning often faces optimization challenges due to the complexity of solving partial differential equations (PDEs), which involve exploring large solution spaces, require numerous iterations, and can lead to unstable training. These challenges arise particularly from the ill-conditioning of the optimization problem caused by the differential terms in the loss function. To address these issues, we propose learning a solver, i.e., solving PDEs using a physics-informed iterative algorithm trained on data. Our method learns to condition a gradient descent algorithm that automatically adapts to each PDE instance, significantly accelerating and stabilizing the optimization process and enabling faster convergence of physics-aware models. Furthermore, while traditional physics-informed methods solve for a single PDE instance, our approach extends to parametric PDEs. Specifically, we integrate the physical loss gradient with PDE parameters, allowing our method to solve over a distribution of PDE parameters, including coefficients, initial conditions, and boundary conditions. We demonstrate the effectiveness of our approach through empirical experiments on multiple datasets, comparing both training and test-time optimization performance. The code is available at https://github.com/2ailesB/neural-parametric-solver.

Learning a Neural Solver for Parametric PDE to Enhance Physics-Informed Methods

TL;DR

This work tackles the optimization bottlenecks of physics-informed PDE solvers by learning a neural optimizer that conditions gradient updates on PDE parameters, enabling rapid convergence for parametric PDEs. By combining a linear basis expansion for the solution with Fourier-based neural conditioning, the approach learns to transform ill-conditioned PDE residual landscapes into well-behaved ones, reducing required iterations from thousands to a handful at test time. Theoretical analysis links the learned solver to preconditioning, showing potential to achieve near-optimal convergence rates under linear assumptions, while extensive experiments demonstrate superior accuracy and fast inference across multiple PDE families. Overall, the method offers a practical path to efficient, parametric PDE solving that complements and often outperforms traditional PINN-based approaches.

Abstract

Physics-informed deep learning often faces optimization challenges due to the complexity of solving partial differential equations (PDEs), which involve exploring large solution spaces, require numerous iterations, and can lead to unstable training. These challenges arise particularly from the ill-conditioning of the optimization problem caused by the differential terms in the loss function. To address these issues, we propose learning a solver, i.e., solving PDEs using a physics-informed iterative algorithm trained on data. Our method learns to condition a gradient descent algorithm that automatically adapts to each PDE instance, significantly accelerating and stabilizing the optimization process and enabling faster convergence of physics-aware models. Furthermore, while traditional physics-informed methods solve for a single PDE instance, our approach extends to parametric PDEs. Specifically, we integrate the physical loss gradient with PDE parameters, allowing our method to solve over a distribution of PDE parameters, including coefficients, initial conditions, and boundary conditions. We demonstrate the effectiveness of our approach through empirical experiments on multiple datasets, comparing both training and test-time optimization performance. The code is available at https://github.com/2ailesB/neural-parametric-solver.

Paper Structure

This paper contains 67 sections, 3 theorems, 54 equations, 41 figures, 18 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Motivation
  3. Poisson equation, 1d.
  4. Approach
  5. Problem statement
  6. Methodology
  7. Training of a physics-informed solver
  8. Theoretical analysis and relation to preconditioning
  9. Experiments
  10. Datasets
  11. Comparison with baselines
  12. Optimization for solving new equations
  13. Computational time
  14. Conclusion
  15. Detailed related Work
  16. ...and 52 more sections

Key Result

Theorem 1

(Convergence rate in the linear case). Given a linear ansatz $u_\Theta(x) = \sum_{i=1}^N \theta_i \phi_i(x)$, assume the conditioner $\mathcal{F}$ behaves like its linearization $P = \text{Jacobian}( \mathcal{F})$, meaning that $\mathcal{F}$ can be replaced by $P$ at any point. Let A be the matrix d Moreover, if $\mathcal{F}$ minimizes $\mathcal{L}_{\text{DATA}}$ this necessarily implies $\kappa(P

Figures (41)

  • Figure 1: Optimization scheme of a physics-informed method with our framework.
  • Figure 2: Test-time optimization based on the physical residual loss $\mathcal{L}_{\textnormal{PDE}}$ for new PDE on Poisson.
  • Figure 3: Samples from the Helmholtz Dataset.
  • Figure 4: Samples from the Poisson Dataset.
  • Figure 5: Samples from the Reaction-Diffusion Dataset.
  • ...and 36 more figures

Theorems & Definitions (6)

  • Theorem 1
  • proof
  • Theorem 2: Convergence rate of PINNs
  • proof
  • Theorem 3
  • proof