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

2024-10-09

International Conference on Learning Representations

AI Summary

Physics-informed neural networks (PINNs) face challenges—including ill-conditioned optimization, slow convergence, and poor generalization—when solving parametric partial differential equations (PDEs). This paper proposes a data-driven neural solver that parameterizes adaptive gradient descent as a neural network, jointly modeling distributions of PDE coefficients and initial/boundary conditions under physical constraints, while dynamically conditioning the optimizer to alleviate loss function ill-conditioning. To our knowledge, this is the first work to introduce neural solvers into parametric PDE settings, enabling end-to-end training via implicit differentiation and backpropagation. Experiments demonstrate a 2–5× speedup in training with enhanced convergence stability. At inference, the solver generalizes robustly to unseen parameter combinations, significantly reducing required iterations while maintaining high accuracy.

Technology Category

• Search and Optimization: Learning to Search
• Constraint Satisfaction and Optimization: Solvers and Tools
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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 addresses parametric PDEs. Specifically, our method integrates the physical loss gradient with the PDE parameters to solve over a distribution of PDE parameters, including coefficients, initial conditions, or boundary conditions. We demonstrate the effectiveness of our method through empirical experiments on multiple datasets, comparing training and test-time optimization performance.

Problem

• Optimizing physics-informed deep learning for complex PDEs
• Addressing unstable training from differential loss terms
• Extending solver to parametric PDEs with varied conditions

Innovation

• Learns solver for parametric PDEs
• Adapts gradient descent per PDE instance
• Integrates physical loss with PDE parameters