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.

Figures (8)

• Figure 1: The SAFE-PIT-CM architecture. A video of physical evolution $V[t]$ is encoded into a latent field $F[t]$; the SAFE operator (frozen PDE solver) propagates the latent field forward in time; a decoder reconstructs the video. The transport coefficient $\hat{\alpha}$ is inferred by an attention-based estimator from the temporal structure of $F$.
• Figure 2: Data flow and training losses. The attention-based estimator predicts $\hat{\alpha}$ from the latent sequence; the frozen SAFE operator sub-steps the PDE forward. Solid red boxes denote unsupervised losses; dotted red boxes denote optional supervised losses available when simulation data is provided.
• Figure 3: Predicted vs. true transport coefficient $\alpha$ on 20 held-out simulations for both diffusion ($\alpha < 0$, blue) and mobility ($\alpha > 0$, red). (a) Inference after training: the model is trained on 40 simulations (sims 0--39) with $N_{\text{sub}} = 50$, then evaluated on 20 held-out test simulations with 50 steps of test-time fine-tuning. MAE $=$ 0.0749, $R^2_{\text{all}} = 0.9245$; diffusion $R^2 = 0.9831$, mobility $R^2 = -0.33$. The trained model recovers diffusion accurately but fails on mobility. (b) Zero-shot inference: a fresh, randomly initialised model is optimised from scratch on each simulation individually (200 steps, no training data). MAE $=$ 0.0117, $R^2_{\text{all}} = 0.9982$; diffusion $R^2 = 1.0000$, mobility $R^2 = 0.9674$.
• Figure 4: Diffusion ($\alpha < 0$): example inference on simulation 40 ($\alpha_{\text{true}} = -0.198$, $\hat{\alpha} = -0.198$). Each row shows (left to right): input field $V[t]$, latent field $F[t]$, reconstructed field $\hat{V}[t]$, and absolute error $|P[t] - F[t+1]|$. The field evolves from a sharp Voronoi microstructure (a) through progressive smoothing (b, c) toward a diffused state (d).
• Figure 5: Mobility ($\alpha > 0$): example inference on simulation 90 ($\alpha_{\text{true}} = +0.198$, $\hat{\alpha} = +0.202$). Same layout as Fig. \ref{['fig:example-diffusion']}. The field sharpens over time: grain boundaries become increasingly pronounced as the reverse heat equation amplifies spatial gradients. Simulations 40 and 90 share the same $|\alpha|$ but opposite sign, illustrating the diffusion/mobility duality.