Examining the robustness of Physics-Informed Neural Networks to noise for Inverse Problems

2025-09-24

AI Summary

This study systematically evaluates the robustness of physics-informed neural networks (PINNs) for solving noisy inverse partial differential equation (PDE) problems, benchmarking them against traditional finite element method (FEM)-based numerical optimization across multiple dimensions. Experiments span varying noise levels, problem dimensions, and data scales, focusing on canonical fluid dynamics inverse problems. Results show that while PINNs underperform conventional methods in low-dimensional, small-data regimes, their accuracy gap narrows substantially in high-dimensional and large-data settings. Moreover, PINNs exhibit intrinsic tolerance to observational noise and circumvent mesh-generation bottlenecks inherent to FEM—particularly in complex geometries and high-dimensional domains. The study further identifies and categorizes three prevalent failure modes in PINN training, providing empirical foundations and concrete directions for designing more robust PINN architectures.

Technology Category

• Machine Learning: Adversarial Learning & Robustness
• Search and Optimization: Sampling/Simulation-based Search
• Intelligent Robots: Learning & Optimization for ROB

Application Category

• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Approximating solutions to partial differential equations (PDEs) is fundamental for the modeling of dynamical systems in science and engineering. Physics-informed neural networks (PINNs) are a recent machine learning-based approach, for which many properties and limitations remain unknown. PINNs are widely accepted as inferior to traditional methods for solving PDEs, such as the finite element method, both with regard to computation time and accuracy. However, PINNs are commonly claimed to show promise in solving inverse problems and handling noisy or incomplete data. We compare the performance of PINNs in solving inverse problems with that of a traditional approach using the finite element method combined with a numerical optimizer. The models are tested on a series of increasingly difficult fluid mechanics problems, with and without noise. We find that while PINNs may require less human effort and specialized knowledge, they are outperformed by the traditional approach. However, the difference appears to decrease with higher dimensions and more data. We identify common failures during training to be addressed if the performance of PINNs on noisy inverse problems is to become more competitive.

Problem

• Evaluating PINNs' robustness to noise in inverse problems
• Comparing PINNs with traditional FEM on fluid mechanics problems
• Identifying training failures to improve PINNs' noisy data performance

Innovation

• PINNs solve inverse problems with noise
• Compared PINNs with finite element method
• Identified training failures for improvement