TRIE: An Evaluation Framework for Stochastic PDE Surrogates

2026-06-30

AI Summary

Existing deterministic neural surrogate models struggle to accurately capture the long-term statistical properties and predictive uncertainty of stochastic partial differential equation (SPDE) systems subject to random perturbations. This work proposes the TRIE evaluation framework—the first comprehensive benchmark tailored for surrogate models of stochastic PDEs—focusing on invariant measure recovery, uncertainty calibration, and efficient probabilistic generation. The authors evaluate multiple approaches, including generative models, Monte Carlo Dropout, and heteroscedastic Gaussian likelihoods, across two chaotic stochastic field SPDEs under various parameter settings. Experimental results demonstrate that the proposed latent generative model consistently achieves the lowest Continuous Ranked Probability Score (CRPS) across all configurations, accurately reproducing the underlying statistical structure while accelerating inference by approximately 12× without compromising long-term statistical fidelity.

Abstract

Many scientific systems exhibit uncertainty from stochastic forcing, unresolved degrees of freedom, or imperfect observations, making reliable surrogate forecasting fundamentally distributional rather than pointwise. For such systems, deterministic neural surrogates fail to capture statistical measures and forecast uncertainty. We introduce TRIE, an evaluation framework for stochastic PDE surrogates that asks whether models reproduce invariant measures, provide trustworthy predictive uncertainty, and scale to efficient probabilistic generation. We demonstrate TRIE on two stationary chaotic spatially extended SPDEs, stochastic Kuramoto--Sivashinsky and stochastic Kolmogorov flow, across 11 parameter values. Our evaluation shows that standard pointwise-trained neural surrogates can produce plausible short rollouts while failing to match long-time statistical structure. Approximate uncertainty methods such as Monte Carlo dropout and heteroscedastic Gaussian likelihoods produce stochastic forecasts, but are often miscalibrated and overconfident under temporal and spatial uncertainty diagnostics. Across these criteria, generative models provide the most consistent performance, accurately capturing invariant measure statistics and achieving the lowest CRPS in all reported probabilistic settings. Finally, we show that latent generative models with automatic dimension discovery retain much of this statistical fidelity while reducing Kolmogorov inference time by roughly $12\times$. We release our code and data at https://github.com/scailab/TRIE-SPDE-Bench to support reproducible evaluation of stochastic PDE forecasting models.

Problem

• stochastic PDE
• surrogate modeling
• predictive uncertainty
• invariant measure
• probabilistic forecasting

Innovation

• stochastic PDE surrogates
• evaluation framework
• invariant measure
• generative models
• predictive uncertainty