SPIEDiff: robust learning of long-time macroscopic dynamics from short-time particle simulations with quantified epistemic uncertainty

TL;DR

SPIEDiff addresses the challenge of learning long-time dissipative macroscopic dynamics from short-time particle data by uniting fluctuation-dissipation theory with conditional diffusion models and epistemic nets. It learns the discretized dissipative operator and free-energy functional in a structure-preserving way, with epinets providing calibrated epistemic uncertainty for both learned components and resulting predictions. The method demonstrates accurate thermodynamics and kinetics discovery across Arrhenius-type systems, including regimes where analytical coarse-grained models fail, and delivers substantial computational savings over direct particle simulations. This yields a robust, scalable pathway for data-driven thermodynamic model discovery with quantified trustworthiness for complex nonequilibrium systems.

Abstract

The data-driven discovery of long-time macroscopic dynamics and thermodynamics of dissipative systems with particle fidelity is hampered by significant obstacles. These include the strong time-scale limitations inherent to particle simulations, the non-uniqueness of the thermodynamic potentials and operators from given macroscopic dynamics, and the need for efficient uncertainty quantification. This paper introduces Statistical-Physics Informed Epistemic Diffusion Models (SPIEDiff), a machine learning framework designed to overcome these limitations in the context of purely dissipative systems by leveraging statistical physics, conditional diffusion models, and epinets. We evaluate the proposed framework on stochastic Arrhenius particle processes and demonstrate that SPIEDiff can accurately uncover both thermodynamics and kinetics, while enabling reliable long-time macroscopic predictions using only short-time particle simulation data. SPIEDiff can deliver accurate predictions with quantified uncertainty in minutes, drastically reducing the computational demand compared to direct particle simulations, which would take days or years in the examples considered. Overall, SPIEDiff offers a robust and trustworthy pathway for the data-driven discovery of thermodynamic models.

Figures (9)

• Figure 1: Schematic overview of the proposed framework for purely dissipative isothermal processes $\partial z/\partial t =-\mathcal{K}_z \delta F/\delta z$ (or analogously, $\partial z/\partial t=\mathcal{M}_z \delta S/\delta z$ in closed systems). Data collected from particle simulations $z_{\epsilon}(x,t)$ serve as inputs, specifically: the coarse-grained field $z(x,t)=\mathbb{E}\left[z_\epsilon\right]$, its short-time evolution $\Delta z(x,t)$, and the discretized dissipative operator entries $\left\langle\gamma_j, \mathcal{K}_z \gamma_i\right\rangle$, using finite element basis $\{\gamma\}$, measured via fluctuation-dissipation relations. These inputs are used in the SPIEDiff (conditional diffusion models augmented with epinets). The framework is trained to learn the underlying thermodynamic structure, i.e., the dissipative operator $\mathcal{K}_z$ and the free energy functional $F[z]$, which are later used to discover the macroscopic evolution equations with quantified epistemic uncertainty.
• Figure 2: SPIEDiff results compared with the analytical long-range model (LRM) and kinetic Monte Carlo (KMC) simulations for the case of long-range interactions: (a) dissipative operator entry $\bar{K}_1$, (b) its epistemic uncertainty, quantified by SPIEDiff, (c) calibrated free energy density $\bar{f}$, and (d-f) macroscopic evolution snapshots at indicated times (CI denotes confidence interval).
• Figure 3: Comparison of the training datasets used for the case of long-range interactions. These are generated from (a) $28$ initial profiles, with $R=10^4$ and (b) the first $4$ initial profiles with $R=10^4$. (c) shows $\Delta \rho_i / \Delta t$ obtained form keeping $28$ initial profiles, but varying $R$ for each profile.
• Figure 4: Comparison of SPIEDiff and baseline models (Stat-PINNs, LRM) under challenging data conditions at selected time instances. (a) and (b) show the results for the scarcer data condition at $t = 0.040$ and $t = 0.060$, respectively, while (c) and (d) present the results for the noisier data condition at the same time instances. For SPIEDiff, both mean predictions and $95 \%$ confidence intervals (CI) are shown.
• Figure 5: Detailed schematic of the SPIEDiff framework architecture and sequential training procedure. The process involves training base conditional DDPMs ($\mathrm{NN}_{K_1}$ and $\mathrm{NN}_{f}$) followed by corresponding epinets ($\mathrm{Epinet}_{K_1}$ and $\mathrm{Epinet}_{f}$) to learn thermodynamic components with quantified epistemic uncertainty. The training order is: (a)$\rightarrow$(b)$\rightarrow$(c).