A Stability-Aware Frozen Euler Autoencoder for Physics-Informed Tracking in Continuum Mechanics (SAFE-PIT-CM)

Abstract

We introduce a Stability-Aware Frozen Euler autoencoder for Physics-Informed Tracking in Continuum Mechanics (SAFE-PIT-CM) that recovers material parameters and temporal field evolution from videos of physical processes. The architecture is an autoencoder whose latent-space transition is governed by a frozen PDE operator: a convolutional encoder maps each frame to a latent field; the SAFE operator propagates it forward via sub-stepped finite differences; and a decoder reconstructs the video. Because the physics is embedded as a frozen, differentiable layer, backpropagation yields gradients that directly supervise an attention-based estimator for the transport coefficient alpha, requiring no ground-truth labels. The SAFE operator is the central contribution. Temporal snapshots are saved at intervals far larger than the simulation time step; a forward Euler step at the frame interval violates the von Neumann stability condition, causing alpha to collapse to an unphysical value. The SAFE operator resolves this by sub-stepping the frozen finite-difference stencil to match the original temporal resolution, restoring stability and enabling accurate parameter recovery. We demonstrate SAFE-PIT-CM on the heat equation (diffusion, alpha < 0) and the reverse heat equation (mobility, alpha > 0). SAFE-PIT-CM also supports zero-shot inference: learning alpha from a single simulation with no training data, using only the SAFE loss as supervision. The zero-shot mode achieves accuracy comparable to a pre-trained model. The architecture generalises to any PDE admitting a convolutional finite-difference discretisation. Because latent dynamics are governed by a known PDE, SAFE-PIT-CM is inherently explainable: every prediction is traceable to a physical transport coefficient and step-by-step PDE propagation.