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

Emil Hovad

cs.LGFeb 27, 2026

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

Authors: Emil Hovad

Organizations: Section for Statistics and Data Analysis, DTU Compute, Technical University of Denmark, Kgs. Lyngby, Denmark

Abstract

Material parameters such as thermal diffusivity govern how microstructural fields evolve during processing, but difficult to measure directly. The Stability-Aware Frozen Euler Physics-Informed Tracking for Continuum Mechanics (SAFE-PIT-CM), is an autoencoder that embeds a frozen convolutional layer as a differentiable PDE solver in its latent-space transition to jointly recover diffusion coefficients and the underlying physical field from temporal observations. When temporal snapshots are saved at intervals coarser than the simulation time step, a single forward Euler step violates the von Neumann stability condition, forcing the learned coefficient to collapse to an unphysical value. Sub-stepping with SAFE restores stability at negligible cost each sub-step is a single frozen convolution, far cheaper than processing more frames with recovery error converging monotonically with substep count. Validated on thermal diffusion in metals, the method recovers both the diffusion coefficient and the physical field with near-perfect accuracy, both with and yet without pre-training. Backpropagation through the frozen operator supervises an attention-based parameter estimator without labelled data. The architecture generalises to any PDE with a convolutional finite-difference discretisation.

Figures & tables

Figure 1: The SAFE-PIT-CM architecture. Input frames $V_{t}$ are separated by $\Delta t_{\text{save}}$ . The encoder maps each frame to a latent field $F_{t}$ ; the SAFE operator propagates $F_{t}$ forward using $N_{\text{sub}}$ sub-steps of size $\Delta t_{\text{sub}}=\Delta t_{\text{save}}/N_{\text{sub}}$ to predict $\hat{F}_{t+1}$ ; the decoder reconstructs $\hat{V}_{t}$ from $F_{t}$ . The diffusion coefficient $\hat{\alpha}$ is inferred by an attention-based estimator from the temporal structure of $\{F_{t}\}$ . Dashed lines indicate the time intervals $\Delta t_{\text{save}}$ and $\Delta t_{\text{sub}}$ .

Component	Description	Parameters
Encoder	3 conv blocks + $1\!\times\!1$ proj.; zero-init	${\sim}7\text{k}$
Physics est.	2 conv + attention ( $d\!=\!32$ , 4 heads) + MLP	${\sim}9\text{k}$
SAFE operator	$N_{\text{sub}}$ frozen Laplacian conv steps	$9$ (frozen)
Decoder	2 conv blocks + $1\!\times\!1$ proj.; zero-init	${\sim}5\text{k}$
Total		${\sim}\textbf{21k}$

Table 1: Network architecture. All convolutions use

3\!\times\!3

kernels with circular padding (periodic boundaries) unless noted. The SAFE operator is frozen (no trainable parameters).

Symbol	Name	Where	Definition	Value
$\Delta t_{\text{sim}}$	Simulation time step	Data generation	(PDE solver step)	$0.005$
$s$	Save interval	Data generation	(steps between frames)	$50$
$\Delta t_{\text{save}}$	Frame interval	Between frames	$s\cdot\Delta t_{\text{sim}}$	$0.25$
$N_{\text{sub}}$	Sub-step count	SAFE operator	(hyperparameter)	$50$ (default)
$\Delta t_{\text{sub}}$	Sub-step size	Within SAFE operator	$\Delta t_{\text{save}}/N_{\text{sub}}$	$0.005$ (at $N_{\text{sub}}\!=\!50$ )

Table 2: Time scales in SAFE-PIT-CM. Two distinct time steps govern the method:

\Delta t_{\text{save}}

, the interval between saved frames in the input data, and

\Delta t_{\text{sub}}

, the internal time step used by the SAFE operator for sub-stepping within each frame pair. The sub-step count

N_{\text{sub}}

is a tuneable hyperparameter; the default is

N_{\text{sub}}=50

(matching the save interval

s

), ablated over

N_{\text{sub}}\in\{2,5,10,25,50\}

in Sect. 4.1

Figure 2: Data flow and training losses. The encoder maps each frame

V_{t}

to a latent field

F_{t}

; the attention-based estimator predicts

\hat{\alpha}

from the sequence

\{F_{t}\}

; the frozen SAFE operator iterates

N_{\text{sub}}

sub-steps (Eq. 6 ) to produce

\hat{F}_{t+1}

; the decoder reconstructs

\hat{V}_{t}

from

F_{t}

. Solid red boxes: unsupervised losses; dotted red boxes: optional supervised losses available when simulation data is provided

Parameter	Symbol	Value
Optimizer		Adam
Learning rate	$\eta$	$1\times 10^{-3}$
Scheduler		ReduceLROnPlateau (0.5, pat. 5)
Early stopping		patience 5
SAFE loss weight	$w_{\text{SAFE}}$	100
Recon. loss weight	$w_{\text{recon}}$	1

Table 3: Training hyperparameters

Setting	MAE α	$\boldsymbol{R^{2}}$
Pure inference	$6.7\times 10^{-1}$	$-1.547$
Pretrained + TTT	$4.1\times 10^{-4}$	$1.000$
Zero-shot TTT	$2.9\times 10^{-5}$	$1.000$

Table 4: Diffusion coefficient recovery on 20 held-out simulations,

\alpha\in[0.03,\,1.56]

Setting	$n$	Mean $\mathcal{L}_{\text{SAFE}}$	Mean $\mathcal{L}_{\text{recon}}$
Pure inference	20	$1.5\times 10^{-6}$	$5.3\times 10^{-5}$
Pretrained + TTT	20	$4.4\times 10^{-10}$	$1.7\times 10^{-7}$
Zero-shot TTT	20	$8.1\times 10^{-11}$	$2.5\times 10^{-7}$

Table 5: Physical field and physics residual quality (

N_{\text{sub}}=50

). The SAFE loss

\mathcal{L}_{\text{SAFE}}

(Eq. 9 ) measures the physics residual in latent space;

\mathcal{L}_{\text{recon}}=\|\hat{V}_{t}-V_{t}\|^{2}

measures the physical field recovery error, since the video frames are direct observations of the field. Both are mean squared errors averaged over all frames and simulations

Panel of Figure 3 — Figure 3: Predicted vs. true diffusion coefficient $\alpha$ on held-out simulations. (a) Pretrained inference: model fine-tuned until convergence per test simulation with $\mathcal{L}_{\text{SAFE}}+\mathcal{L}_{\text{recon}}$ (all learnable parameters updated; SAFE operator frozen). (b) Test-time training: fresh model optimised until convergence per simulation with $\mathcal{L}_{\text{SAFE}}$ alone (no pre-training required)

Panel of Figure 4 — Figure 4: Example inference on a held-out simulation. Each row shows (left to right): input field $V_{t}$ , latent field $F_{t}$ , reconstructed field $\hat{V}_{t}$ , and absolute error $|\hat{F}_{t+1}-F_{t+1}|$ . The field evolves from a sharp Voronoi microstructure (a) through progressive smoothing (b, c) toward a diffused state (d)

Figure 5: Convergence of

\alpha

recovery with sub-step count

N_{\text{sub}}

(zero-shot TTT). Left: MAE (log scale). Right:

R^{2}

. Recovery collapses at

N_{\text{sub}}\leq 2

due to the stability violation; performance converges monotonically with increasing

N_{\text{sub}}

$\boldsymbol{N_{\text{sub}}}$	$\boldsymbol{\Delta t_{\text{sub}}}$	$\boldsymbol{\sigma_{\max}}$	$\boldsymbol{\alpha_{\max}}$	Stable?	$\mathrm{MAE}_{\alpha}$	$\boldsymbol{R^{2}}$	$\mathcal{L}_{\text{SAFE}}$
1	0.2500	13.60	0.25	No	0.5042	$-0.5242680$	$6.0\times 10^{-6}$
2	0.1250	6.80	0.50	No	0.3449	$0.1385034$	$3.0\times 10^{-6}$
5	0.0500	2.72	1.25	No	0.0798	$0.9468073$	$1.0\times 10^{-7}$
10	0.0250	1.36	2.50	Yes	0.0153	$0.9987317$	$1.0\times 10^{-8}$
25	0.0100	0.54	6.25	Yes	0.0048	$0.9998469$	$1.0\times 10^{-8}$
50	0.0050	0.27	12.50	Yes	0.0006	$0.9999926$	$3.0\times 10^{-11}$

Table 6: Stability properties and zero-shot

\alpha

recovery for each tested sub-step count

N_{\text{sub}}

. The von Neumann stability number

\sigma=\Delta t_{\text{sub}}\cdot\alpha\cdot\lambda_{\max}

must satisfy

\sigma<2

; increasing

N_{\text{sub}}

reduces

\Delta t_{\text{sub}}

and expands the recoverable

\alpha

range. The data spans

\alpha\in[0.01,1.7]

;

\sigma_{\max}

is evaluated at

\alpha=1.7

. The last three columns report zero-shot TTT recovery quality on 20 held-out simulations.

\mathcal{L}_{\text{recon}}

is zero to machine precision in all cases because the encoder and decoder are frozen at their zero-init state (see also Fig. 5 )

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Parameter	Symbol	Value
Grid size	$N_{x}\times N_{y}$	$128\times 128$
Grid spacing	$\Delta x=\Delta y$	$0.5$
Time step	$\Delta t_{\text{sim}}$	$0.005$
Number of time steps	$N_{\text{step}}$	$5000$
Save interval	$s$	$50$
Saved frame interval	$\Delta t_{\text{save}}$	$0.25$

Appendix

Table 7: Data generation parameters

Scenario	$\boldsymbol{\Delta t_{\text{sub}}}$	$\boldsymbol{\sigma}$ at $\alpha=1.7$	$\boldsymbol{\alpha_{\max}}$	Stable?
No sub-stepping ( $N_{\text{sub}}=1$ )	$0.25$	$13.6$	$0.25$	No
Sub-stepping ( $N_{\text{sub}}=50$ )	$0.005$	$0.27$	$12.5$	Yes

Appendix

Table 8: Von Neumann stability for the SAFE-PIT-CM grid (

\Delta x=\Delta y=0.5

\lambda_{\max}=32

). The stability condition is

\sigma=\Delta t_{\text{sub}}\cdot\alpha\cdot\lambda_{\max}<2

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