PhysSAE reveals hidden physical features inside neural networks for physics problems
PhysSAE: Mechanistic Interpretability with Sparse Autoencoders
Machine Learning
Summary
Physics-Informed Neural Networks (PINNs) are used to solve physics problems by embedding physical equations into training, but it wasn't clear what these networks actually learn inside their hidden layers. The authors introduce PhysSAE, a method that uses special sparse autoencoders to study these hidden parts and link them directly to known physical quantities. They show that the features found by PhysSAE match real physical concepts very well and can be tested to see how important they are inside the network. This makes it easier to understand how PINNs represent physics and could help make these AI tools more transparent and trustworthy.
Physics-Informed Neural NetworksPartial Differential EquationsSparse AutoencoderLatent RepresentationsMechanistic InterpretabilityCausal InterventionPearson CorrelationPrincipal Component AnalysisIndependent Component Analysis
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.