GIMLET: Generalizable and Interpretable Model Learning through Embedded Thermodynamics

TL;DR

GIMLET presents a library-free framework for gray-box discovery of constitutive closures in flow and transport PDEs by embedding thermodynamics through a variational principle. Unknown terms are expressed as functional derivatives of learned free-energy and dissipation functionals, parameterized by INN and CINN networks and trained with a physics-informed loss via automatic differentiation. The method is validated on Burgers, Kuramoto–Sivashinsky, and Navier–Stokes systems (both Newtonian and non-Newtonian), showing accurate term discovery and strong generalization to new datasets governed by the same equations. The thermodynamic structure provides direct physical interpretability, while obviating reliance on predefined function libraries, enabling transferable closures across regimes and potential extensions to broader multiphysics problems.

Abstract

We develop a data-driven framework for discovering constitutive relations in models of fluid flow and scalar transport. Our approach infers unknown closure terms in the governing equations (gray-box discovery) under the assumption that the temporal derivative, convective transport, and pressure-gradient contributions are known. The formulation is rooted in a variational principle from nonequilibrium thermodynamics, where the dynamics is defined by a free-energy functional and a dissipation functional. The unknown constitutive terms arise as functional derivatives of these functionals with respect to the state variables. To enable a flexible and structured model discovery, the free-energy and dissipation functionals are parameterized using neural networks, while their functional derivatives are obtained via automatic differentiation. This construction enforces thermodynamic consistency by design, ensuring monotonic decay of the total free energy and non-negative entropy production. The resulting method, termed GIMLET (Generalizable and Interpretable Model Learning through Embedded Thermodynamics), avoids reliance on a predefined library of candidate functions, unlike sparse regression or symbolic identification approaches. The learned models are generalizable in that functionals identified from one dataset can be transferred to distinct datasets governed by the same underlying equations. Moreover, the inferred free-energy and dissipation functions provide direct physical interpretability of the learned dynamics. The framework is demonstrated on several benchmark systems, including the viscous Burgers equation, the Kuramoto--Sivashinsky equation, and the incompressible Navier--Stokes equations for both Newtonian and non-Newtonian fluids.

Figures (11)

• Figure 1: Schematic architecture of the GIMLET for data-driven discovery of constitutive models in flow and transport PDEs.
• Figure 2: Results of model discovery and its application for Burger's equation. Upper panels are for model discovery, and lower panels are for model application. (a,f): history of loss function, where the total value, $\mathcal{J}_p$ (GE), $\mathcal{J}_d$ (Data), and the test loss (Test) are separately plotted. (b,g): Spatio-temporal distribution of solution calculated by GIMLET. (c,h): Solutions by the spectral method. (d,i): Differences between predictions and data. (e,j): Selected snapshots.
• Figure 3: (a) Dissipation function $\theta(u_x)$, where the green solid line indicates the function discovered by the GIMLET and the orange dashed line the true function. (b,c) The histograms of $u_x$ in the datasets used for discovery and application, respectively. (d) Relation between trained $\theta$ and $\eta/2 \left( u_x \right)^2$, where the blue circles are values in the dataset A.
• Figure 4: Results of model training for Kuramoto-Sivashinsky equation. Upper panels are for model discovery, and lower panels are for model application. (a,f): History of loss function, where the total value, $\mathcal{J}_p$ (GE), $\mathcal{J}_d$ (Data), and the test loss (Test) are separately plotted. (b,g): Spatio-temporal distribution of solution calculated by GIMLET. (c,h): Solutions by the spectral method, where the time regimes shown in half-transparent are not used for training. (d,i): Differences between predictions and data. (e,j): Selected snapshots.
• Figure 5: (a,d): Free energy functions $g_0\left( \phi \right)$ and $g_1\left( \phi_x \right)$, where the green solid lines indicate the functions discovered by the GIMLET and the orange dashed lines the true functions. (b,e): The histograms of $\phi$ and $\phi_x$ in the dataset A used for discovery. (c,f): The histograms in the dataset B for the generalization check. (g): Relation between trained $g_0$ and $-\beta/2 \left( \phi \right)^2$, where the blue circles are values in the dataset A. (h): Relation between $g_1$ and $\gamma/2 \left( \phi_x \right)^2$.