Newton Informed Neural Operator for Computing Multiple Solutions of Nonlinear Partials Differential Equations

TL;DR

The paper tackles the challenge of solving nonlinear PDEs that admit multiple solutions, where traditional neural solvers often fail to capture all branches. It introduces the Newton Informed Neural Operator (NINO), which learns the Newton map by approximating the linearized step with a neural operator, enabling local well-posedness and simultaneous discovery of multiple solutions with reduced supervised data. The authors develop a principled loss framework that combines a supervised mean-squared error term with a Newton-based residual term, along with theoretical guidance on approximation and generalization to support data-efficient training. Experiments on convex and non-convex problems, plus the Gray-Scott model, show that NINO achieves improved generalization, learns multiple solution branches from limited data, and offers substantial speedups over classical Newton solvers when deployed on GPUs. This approach has practical impact for physics, chemistry, and biology where multiplicity in PDE solutions is a fundamental feature and computational efficiency is critical.

Abstract

Solving nonlinear partial differential equations (PDEs) with multiple solutions using neural networks has found widespread applications in various fields such as physics, biology, and engineering. However, classical neural network methods for solving nonlinear PDEs, such as Physics-Informed Neural Networks (PINN), Deep Ritz methods, and DeepONet, often encounter challenges when confronted with the presence of multiple solutions inherent in the nonlinear problem. These methods may encounter ill-posedness issues. In this paper, we propose a novel approach called the Newton Informed Neural Operator, which builds upon existing neural network techniques to tackle nonlinearities. Our method combines classical Newton methods, addressing well-posed problems, and efficiently learns multiple solutions in a single learning process while requiring fewer supervised data points compared to existing neural network methods.

Figures (7)

• Figure 1: Training and testing performance of DeepONet under various conditions. For the detailed experiments settings, please refer to the Appendix \ref{['sec:experiment_settings']}.
• Figure 2: Solutions of 2D Non-convex problem \ref{['2dex']}
• Figure 3: The convergence behavior of the Neural Operator-based solver.
• Figure 4: Examples of steady states of the Gray Scott model
• Figure 5: Three examples depicting the evolution from the initial state to the steady state via Newton's method.

Theorems & Definitions (12)

• Remark 1
• Theorem 1
• Theorem 2
• Remark 2
• Corollary 1
• Remark 3
• Definition 1: Sobolev Spaces evans2022partial
• Proposition 1: mhaskar1996neural
• Proposition 2: poggio2017andyang2023nearly
• Lemma 1: lanthaler2022error