naPINN: Noise-Adaptive Physics-Informed Neural Networks for Recovering Physics from Corrupted Measurement

TL;DR

This work tackles the fragility of Physics-Informed Neural Networks (PINNs) to non-Gaussian measurement noise and gross outliers in inverse PDE problems. It introduces naPINN, a noise-adaptive PINN that couples a one-dimensional Energy-Based Model (EBM) to learn the empirical distribution of prediction residuals and a trainable reliability gate to adaptively weight data points, with a rejection-cost regularizer to prevent degenerate data rejection. The training proceeds in staged phases—PINN warm-up, EBM initialization, and joint optimization with a running-statistics normalization—and employs an energy-guided gating mechanism to downweight unreliable measurements. Experimental results on 2D Burgers’, Allen–Cahn, and lambda–omega reaction–diffusion systems show that naPINN significantly outperforms robust PINN baselines under severe data corruption, while simultaneously providing interpretable insights into noise structure and outlier behavior. The approach advances robust scientific machine learning by enabling accurate physics recovery in the presence of unknown, multimodal noise and sensor faults, with practical implications for data-driven PDE inference in real-world sensing settings.

Abstract

Physics-Informed Neural Networks (PINNs) are effective methods for solving inverse problems and discovering governing equations from observational data. However, their performance degrades significantly under complex measurement noise and gross outliers. To address this issue, we propose the Noise-Adaptive Physics-Informed Neural Network (naPINN), which robustly recovers physical solutions from corrupted measurements without prior knowledge of the noise distribution. naPINN embeds an energy-based model into the training loop to learn the latent distribution of prediction residuals. Leveraging the learned energy landscape, a trainable reliability gate adaptively filters data points exhibiting high energy, while a rejection cost regularization prevents trivial solutions where valid data are discarded. We demonstrate the efficacy of naPINN on various benchmark partial differential equations corrupted by non-Gaussian noise and varying rates of outliers. The results show that naPINN significantly outperforms existing robust PINN baselines, successfully isolating outliers and accurately reconstructing the dynamics under severe data corruption.

Figures (7)

• Figure 1: Overview of the naPINN framework. Measurement residuals are modeled by an energy-based model (EBM) to estimate their log-likelihood under the learned noise distribution. A trainable reliability gate converts this information into adaptive weights, selectively filtering unreliable measurements when computing the data loss.
• Figure 2: Qualitative comparison of solution reconstruction under corrupted measurements. For each PDE benchmark, we visualize the ground-truth solution, predictions from naPINN and a standard PINN, and their corresponding absolute errors at two representative time steps. Different colormaps are used across benchmarks to enhance visibility.
• Figure 3: Robustness comparison under increasing outlier ratios on three PDE benchmarks. The relative RMSE (rMSE) is plotted as a function of the outlier ratio, with error bars denoting one standard deviation over independent runs.
• Figure 4: Comparison of residual distributions learned by the EBM after EBM initialization and after joint naPINN training. Joint optimization enables the EBM to accurately approximate the underlying noise distribution.
• Figure 5: Visualization of the trained reliability gate. Normal and outlier measurement points are projected onto the learned gating function, illustrating adaptive filtering of unlikely residuals.