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Conditional Denoising Model as a Physical Surrogate Model

José Afonso, Pedro Viegas, Rodrigo Ventura, Vasco Guerra

TL;DR

This work tackles surrogate modeling for complex physical systems under data scarcity, where traditional physics-informed losses may fail to ensure strict adherence to governing equations. It introduces the Conditional Denoising Model (CDM), a time-independent (CDM-0) or time-conditioned (CDM-t) denoiser that learns a vector field projecting noisy states onto the physical solution manifold, translating inference into a deterministic fixed-point refinement or a generative flow. The training leverages a continuous noise schedule and a denoising objective that connects to score matching and ELBO, enabling strong implicit regularization without explicit equation-based losses. Empirical results on a LoKI low-temperature plasma benchmark show that CDM achieves higher data and parameter efficiency than physics-consistent baselines while reducing constraint violations, illustrating the practicality of learning physical manifold geometry for data-scarce scientific domains.

Abstract

Surrogate modeling for complex physical systems typically faces a trade-off between data-fitting accuracy and physical consistency. Physics-consistent approaches typically treat physical laws as soft constraints within the loss function, a strategy that frequently fails to guarantee strict adherence to the governing equations, or rely on post-processing corrections that do not intrinsically learn the underlying solution geometry. To address these limitations, we introduce the {Conditional Denoising Model (CDM)}, a generative model designed to learn the geometry of the physical manifold itself. By training the network to restore clean states from noisy ones, the model learns a vector field that points continuously towards the valid solution subspace. We introduce a time-independent formulation that transforms inference into a deterministic fixed-point iteration, effectively projecting noisy approximations onto the equilibrium manifold. Validated on a low-temperature plasma physics and chemistry benchmark, the CDM achieves higher parameter and data efficiency than physics-consistent baselines. Crucially, we demonstrate that the denoising objective acts as a powerful implicit regularizer: despite never seeing the governing equations during training, the model adheres to physical constraints more strictly than baselines trained with explicit physics losses.

Conditional Denoising Model as a Physical Surrogate Model

TL;DR

This work tackles surrogate modeling for complex physical systems under data scarcity, where traditional physics-informed losses may fail to ensure strict adherence to governing equations. It introduces the Conditional Denoising Model (CDM), a time-independent (CDM-0) or time-conditioned (CDM-t) denoiser that learns a vector field projecting noisy states onto the physical solution manifold, translating inference into a deterministic fixed-point refinement or a generative flow. The training leverages a continuous noise schedule and a denoising objective that connects to score matching and ELBO, enabling strong implicit regularization without explicit equation-based losses. Empirical results on a LoKI low-temperature plasma benchmark show that CDM achieves higher data and parameter efficiency than physics-consistent baselines while reducing constraint violations, illustrating the practicality of learning physical manifold geometry for data-scarce scientific domains.

Abstract

Surrogate modeling for complex physical systems typically faces a trade-off between data-fitting accuracy and physical consistency. Physics-consistent approaches typically treat physical laws as soft constraints within the loss function, a strategy that frequently fails to guarantee strict adherence to the governing equations, or rely on post-processing corrections that do not intrinsically learn the underlying solution geometry. To address these limitations, we introduce the {Conditional Denoising Model (CDM)}, a generative model designed to learn the geometry of the physical manifold itself. By training the network to restore clean states from noisy ones, the model learns a vector field that points continuously towards the valid solution subspace. We introduce a time-independent formulation that transforms inference into a deterministic fixed-point iteration, effectively projecting noisy approximations onto the equilibrium manifold. Validated on a low-temperature plasma physics and chemistry benchmark, the CDM achieves higher parameter and data efficiency than physics-consistent baselines. Crucially, we demonstrate that the denoising objective acts as a powerful implicit regularizer: despite never seeing the governing equations during training, the model adheres to physical constraints more strictly than baselines trained with explicit physics losses.
Paper Structure (23 sections, 22 equations, 3 figures, 2 tables, 3 algorithms)

This paper contains 23 sections, 22 equations, 3 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Generative Modeling via Denoising
  4. The Physical Manifold Assumption
  5. Manifold Learning via Denoising Autoencoders
  6. Methodology: Conditional Denoising Model (CDM)
  7. Training Objective: Multi-Scale Manifold Learning
  8. Theoretical Foundations
  9. Inference as a Generative Flow
  10. Derivation of the Inference ODE
  11. Discrete Sampling Algorithm
  12. Architectural Variants: Time-Dependency
  13. Explicit Time-Dependent Vector Field (CDM-$t$)
  14. Implicit Time-Independent Vector Field (CDM-$0$)
  15. Justification of the Denoising Objective
  16. ...and 8 more sections

Figures (3)

  • Figure 1: Performance comparison between Physics-Consistent Baselines and Conditional Denoising Models (CDM). The top row (a, b) reports the predictive accuracy (Test RMSE), while the bottom row (c, d) evaluates physical consistency (Physics RMSE, measuring constraint violations). The left column (a, c) analyzes the impact of model complexity (number of parameters), while the right column (b, d) evaluates robustness to data scarcity (training set fraction). Shaded regions indicate the standard deviation across $10$ independent training runs. For the data-scarcity plots, the CDMs and baselines models have $\sim 15000$. Shaded regions represent the standard deviation across 10 distinct train-test-validation splits.
  • Figure 2: Ablation study on noise scaling and inference convergence. (a) Impact of the training noise scale $\sigma_{\max}$ on test accuracy (b) Convergence profile of time-independent CDM-$0$ models; the adaptive step size accelerates early-stage convergence, with both variants saturating around $50$--$80$ iterations. (c, d) Analysis of CDM-$t$ inference parameters. Panel (c) shows the effect of schedule resolution $T$ (with fixed $K=1$), saturating at $T \approx 30$. Panel (d) shows the impact of refinement steps $K$ (using a sparse schedule $T=3$). Shaded regions represent the standard deviation across 10 distinct train-test-validation splits. Note the logarithmic scale on the y-axis for Panels (b-c).
  • Figure : CDM Training