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 Vt are separated by Δtsave . The encoder maps each frame to a latent field Ft ; the SAFE operator propagates Ft forward using Nsub sub-steps of size Δtsub=Δtsave/Nsub to predict F^t+1 ; the decoder reconstructs V^t from Ft . The diffusion coefficient α^ is inferred by an attention-based estimator from the temporal structure of {Ft} . Dashed lines indicate the time intervals Δtsave and Δtsub .
Component
Description
Parameters
Encoder
3 conv blocks + 1×1 proj.; zero-init
∼7k
Physics est.
2 conv + attention ( d=32 , 4 heads) + MLP
∼9k
SAFE operator
Nsub frozen Laplacian conv steps
9 (frozen)
Decoder
2 conv blocks + 1×1 proj.; zero-init
∼5k
Total
∼21k
Table 1: Network architecture. All convolutions use 3×3 kernels with circular padding (periodic boundaries) unless noted. The SAFE operator is frozen (no trainable parameters).
Symbol
Name
Where
Definition
Value
Δtsim
Simulation time step
Data generation
(PDE solver step)
0.005
s
Save interval
Data generation
(steps between frames)
50
Δtsave
Frame interval
Between frames
s⋅Δtsim
0.25
Nsub
Sub-step count
SAFE operator
(hyperparameter)
50 (default)
Δtsub
Sub-step size
Within SAFE operator
Δtsave/Nsub
0.005 (at Nsub=50 )
Table 2: Time scales in SAFE-PIT-CM. Two distinct time steps govern the method: Δtsave , the interval between saved frames in the input data, and Δtsub , the internal time step used by the SAFE operator for sub-stepping within each frame pair. The sub-step count Nsub is a tuneable hyperparameter; the default is Nsub=50 (matching the save interval s ), ablated over Nsub∈{2,5,10,25,50} in Sect. 4.1
Figure 2: Data flow and training losses. The encoder maps each frame Vt to a latent field Ft ; the attention-based estimator predicts α^ from the sequence {Ft} ; the frozen SAFE operator iterates Nsub sub-steps (Eq. 6 ) to produce F^t+1 ; the decoder reconstructs V^t from Ft . Solid red boxes: unsupervised losses; dotted red boxes: optional supervised losses available when simulation data is provided
Parameter
Symbol
Value
Optimizer
Adam
Learning rate
η
1×10−3
Scheduler
ReduceLROnPlateau (0.5, pat. 5)
Early stopping
patience 5
SAFE loss weight
wSAFE
100
Recon. loss weight
wrecon
1
Table 3: Training hyperparameters
Setting
MAE α
R2
Pure inference
6.7×10−1
−1.547
Pretrained + TTT
4.1×10−4
1.000
Zero-shot TTT
2.9×10−5
1.000
Table 4: Diffusion coefficient recovery on 20 held-out simulations, α∈[0.03,1.56]
Setting
n
Mean LSAFE
Mean Lrecon
Pure inference
20
1.5×10−6
5.3×10−5
Pretrained + TTT
20
4.4×10−10
1.7×10−7
Zero-shot TTT
20
8.1×10−11
2.5×10−7
Table 5: Physical field and physics residual quality ( Nsub=50 ). The SAFE loss LSAFE (Eq. 9 ) measures the physics residual in latent space; Lrecon=∥V^t−Vt∥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
Figure 3: Predicted vs. true diffusion coefficient α on held-out simulations. (a) Pretrained inference: model fine-tuned until convergence per test simulation with LSAFE+Lrecon (all learnable parameters updated; SAFE operator frozen). (b) Test-time training: fresh model optimised until convergence per simulation with LSAFE alone (no pre-training required)
Figure 4: Example inference on a held-out simulation. Each row shows (left to right): input field Vt , latent field Ft , reconstructed field V^t , and absolute error ∣F^t+1−Ft+1∣ . The field evolves from a sharp Voronoi microstructure (a) through progressive smoothing (b, c) toward a diffused state (d)
Figure 5: Convergence of α recovery with sub-step count Nsub (zero-shot TTT). Left: MAE (log scale). Right: R2 . Recovery collapses at Nsub≤2 due to the stability violation; performance converges monotonically with increasing Nsub
Nsub
Δtsub
σmax
αmax
Stable?
MAEα
R2
LSAFE
1
0.2500
13.60
0.25
No
0.5042
−0.5242680
6.0×10−6
2
0.1250
6.80
0.50
No
0.3449
0.1385034
3.0×10−6
5
0.0500
2.72
1.25
No
0.0798
0.9468073
1.0×10−7
10
0.0250
1.36
2.50
Yes
0.0153
0.9987317
1.0×10−8
25
0.0100
0.54
6.25
Yes
0.0048
0.9998469
1.0×10−8
50
0.0050
0.27
12.50
Yes
0.0006
0.9999926
3.0×10−11
Table 6: Stability properties and zero-shot α recovery for each tested sub-step count Nsub . The von Neumann stability number σ=Δtsub⋅α⋅λmax must satisfy σ<2 ; increasing Nsub reduces Δtsub and expands the recoverable α range. The data spans α∈[0.01,1.7] ; σmax is evaluated at α=1.7 . The last three columns report zero-shot TTT recovery quality on 20 held-out simulations. Lrecon 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
Nx×Ny
128×128
Grid spacing
Δx=Δy
0.5
Time step
Δtsim
0.005
Number of time steps
Nstep
5000
Save interval
s
50
Saved frame interval
Δtsave
0.25
Appendix
Table 7: Data generation parameters
Scenario
Δtsub
σ at α=1.7
αmax
Stable?
No sub-stepping ( Nsub=1 )
0.25
13.6
0.25
No
Sub-stepping ( Nsub=50 )
0.005
0.27
12.5
Yes
Appendix
Table 8: Von Neumann stability for the SAFE-PIT-CM grid ( Δx=Δy=0.5 , λmax=32 ). The stability condition is σ=Δtsub⋅α⋅λmax<2
We propose a physics-informed consistency modeling framework for solving partial differential equations (PDEs) via fast, few-step generative inference. We identify a key stability challenge in physics-constrained consistency training, where PDE residuals can drive the model toward trivial or degenerate solutions, degrading the learned data distribution. To address this, we introduce a structure-preserving two-stage training strategy that decouples distribution learning from physics enforcement by freezing the coefficient decoder during physics-informed fine-tuning. We further propose a two-step residual objective that enforces physical consistency on refined, structurally valid generative trajectories rather than noisy single-step predictions. The resulting framework enables stable, high-fidelity inference for both unconditional generation and forward problems. We demonstrate that forward solutions can be obtained via a projection-based zero-shot inpainting procedure, achieving consistent accuracy of diffusion baselines with orders of magnitude reduction in computational cost.
Che-Chia Chang, Chen-Yang Dai, Te-Sheng Lin +2
Institute of Artificial Intelligence Innovation National Yang Ming Chiao Tung University Hsinchu, Taiwan · Department of Applied Mathematics National Yang Ming Chiao Tung University Hsinchu, Taiwan · Department of Applied Mathematics National Yang Ming Chiao Tung University Hsinchu, Taiwan National Center for Theoretical Sciences National Taiwan University Taipei, Taiwan
Recovering continuous-time dynamics from discrete observations is difficult because local supervision (e.g., pointwise regression targets, derivative approximations, or equation residuals) loses fidelity as the observation interval grows. We replace local supervision with a global structural constraint: any flow representing autonomous dynamics must satisfy the semi-group property under time translation. We train a time-conditioned secant velocity field whose deviation from this property, which we call Symmetry Rupture, serves two purposes. As a training regularizer, it confines the hypothesis space to flows that compose consistently across temporal scales. As an inference oracle, it lets the solver select the largest step size that preserves internal consistency, replacing the local truncation error that conventional adaptive solvers depend on. On the diffusion-reaction benchmark under time-informed inference, our method reduces rollout RMSE by 87% while using 5x fewer function evaluations than a Neural ODE baseline. In the more demanding direct auto-regressive setting, where the model must predict distant future frames without intermediate temporal cues, our adaptive solver allocates compute based on local geometric complexity -- maintaining the lowest rollout RMSE on two of three PDE benchmarks while baselines either diverge or require up to an order of magnitude more function evaluations to remain stable.
Yuxiang Luo, Andrew Perrault
Department of Computer Science and Engineering The Ohio State University
Conditional Diffusion Models are powerful surrogates for emulating complex spatiotemporal dynamics, yet they often fail to match the accuracy of deterministic neural emulators for high-precision tasks. In this work, we address two critical limitations of autoregressive PDE diffusion models: their sub-optimal single-step accuracy and the prohibitive computational cost of unrolled training. First, we characterize the relationship between the noise schedule, the reconstruction error reduction rate and the diffusion exposure bias, demonstrating that standard schedules lead to suboptimal reconstruction error. Leveraging this insight, we propose an Adaptive Noise Schedule framework that minimizes inference reconstruction error by dynamically constraining the model's exposure bias. We further show that this optimized schedule enables a fast Proxy Unrolled Training method to stabilize long-term rollouts without the cost of full Markov Chain sampling. Both proposed methods enable significant improvements in short-term accuracy and long-term stability over diffusion and deterministic baselines on diverse benchmarks, including forced Navier-Stokes, Kuramoto-Sivashinsky and Transonic Flow.