Unlearning Noise in PINNs: A Selective Pruning Framework for PDE Inverse Problems

TL;DR

P-DEPRECATED

Abstract

Physics-informed neural networks (PINNs) provide a promising framework for solving inverse problems governed by partial differential equations (PDEs) by integrating observational data and physical constraints in a unified optimization objective. However, the ill-posed nature of PDE inverse problems makes them highly sensitive to noise. Even a small fraction of corrupted observations can distort internal neural representations, severely impairing accuracy and destabilizing training. Motivated by recent advances in machine unlearning and structured network pruning, we propose P-PINN, a selective pruning framework designed to unlearn the influence of corrupted data in a pretrained PINN. Specifically, starting from a PINN trained on the full dataset, P-PINN evaluates a joint residual--data fidelity indicator, a weighted combination of data misfit and PDE residuals, to partition the training set into reliable and corrupted subsets. Next, we introduce a bias-based neuron importance measure that quantifies directional activation discrepancies between the two subsets, identifying neurons whose representations are predominantly driven by corrupted samples. Building on this, an iterative pruning strategy then removes noise-sensitive neurons layer by layer. The resulting pruned network is fine-tuned on the reliable data subject to the original PDE constraints, acting as a lightweight post-processing stage rather than a complete retraining. Numerical experiments on extensive PDE inverse-problem benchmarks demonstrate that P-PINN substantially improves robustness, accuracy, and training stability under noisy conditions, achieving up to a 96.6\% reduction in relative error compared with baseline PINNs. These results indicate that activation-level post hoc pruning is a promising mechanism for enhancing the reliability of physics-informed learning in noise-contaminated settings.

Figures (9)

• Figure 1: Schematic representation of the PINN formulation used in this work.
• Figure 1: P-PINN workflow and a Navier--Stokes inverse example.Left: schematic of the P-PINN pipeline. Right: Navier--Stokes inversion: A baseline PINN is trained on noisy space--time observations; the observations are then partitioned into $\mathcal{D}_{\text{retain}}$ (blue) and $\mathcal{D}_{\text{forget}}$ (red) using residual--data-fidelity scores. The network is iteratively pruned and selectively fine-tuned on $\mathcal{D}_{\text{retain}}$ to obtain P-PINN. Color maps compare the baseline and P-PINN reconstructions of the pressure field $p(x,y,t{=}1)$ with the reference solution.
• Figure 1: Solution fields for the one-dimensional wave data assimilation problem: (a) standard PINN prediction, which roughly identifies the correct locations of the wave lobes but is contaminated by pronounced spurious oscillations; (b) exact solution $u(x,t)$; (c) P-PINN prediction after selective pruning and fine-tuning, which closely matches the smooth standing-wave structure of the exact solution.
• Figure 2: Heat equation data assimilation: (a) standard PINN prediction; (b) exact solution $u(x,t)$; (c) P-PINN prediction after selective pruning and fine-tuning.
• Figure 3: Poisson data assimilation: (a) standard PINN prediction; (b) exact solution $u(x,y)$; (c) P-PINN prediction.