PhysSAE: Mechanistic Interpretability of PINNs with Sparse Autoencoders

Abstract

Physics-Informed Neural Networks (PINNs) embed PDE residuals into neural network training, but their internal representations remain opaque: it is unknown what physical features their hidden layers encode or whether those features have a localized causal role. We present PhysSAE, a mechanistic interpretability framework that trains overcomplete sparse autoencoders (SAEs) on PINN penultimate-layer activations and evaluates dictionary atoms through direct causal intervention in the original frozen hidden state: $h_{\mathrm{cf}} = h - \alpha z_k d_k$, bypassing the SAE decoder entirely. Across six PDE families, with 3 PINN seeds and 3 SAE seeds each---we show that (i) Our discovered SAE atoms align with independently-defined physical observables (max Pearson $|r|=0.951$, always $\gg$ permutation null), (ii) the causal footprint of top-aligned atom ablation is 1.2--4.2$\times$ more spatially concentrated canonical than PCA or ICA interventions, and (iii) top-aligned atoms outperform matched random controls on causal localization for structured physical concepts (ESF$_{80}$ advantage 0.04-0.44). Two-atom bilateral representations improve concept regression R$^2$ by $\Delta R^2\!=\!0.05\text{-}0.15$ over single atoms, while random pairs decrease it by up to 0.60. These results demonstrate that PINNs develop sparse, physically structured latent representations that can be identified and causally interrogated post-hoc, opening a path toward interpretability-aware scientific machine learning.