PhysSAE reveals hidden physical features in neural networks modeling PDEs
PhysSAE: Mechanistic Interpretability with Sparse Autoencoders
Machine Learning
Summary
Physics-informed neural networks (PINNs) help solve complex physical equations but it's unclear what their hidden layers are actually learning. The authors developed PhysSAE, a way to find meaningful, sparse building blocks inside these networks and test how important each is to the network’s understanding of physical concepts. They showed these building blocks match known physical properties and have a clear effect when altered, making these networks easier to interpret. This approach helps scientists trust and understand AI models that study physical phenomena.
What this means in practice
- •For engineering simulation teams: Improve interpretability of AI models solving PDEs by identifying causal physical features learned inside neural networks.
- •For machine learning model auditors: Enable targeted intervention on neural network components to verify and validate learned physical concepts in scientific modeling.
Authors
Nandita N. Patil, Eshwar R. A., Gajanan V. Honnavar
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: $h_{\mathrm{cf}} = h - αz_k d_k$, 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 $ΔR^2\!=\!0.05\text{-}0.15$ 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.