Differentiable Thermodynamic Phase-Equilibria for Machine Learning

Karim K. Ben Hicham; Moreno Ascani; Jan G. Rittig; Alexander Mitsos

cs.LGMar 11, 2026

Differentiable Thermodynamic Phase-Equilibria for Machine Learning

Authors: Karim K. Ben Hicham, Moreno Ascani, Jan G. Rittig, Alexander Mitsos

Organizations: RWTH Aachen University, Process Systems Engineering (AVT.SVT), 52074 Aachen, Germany · Forschungszentrum Jülich GmbH, Institute of Climate and Energy Systems ICE-1: Energy Systems Engineering, 52425 Jülich, Germany · JARA-ENERGY, 52056 Aachen, Germany

Abstract

Accurate prediction of phase equilibria remains a central challenge in chemical engineering. Physics-consistent machine learning methods that incorporate thermodynamic structure into neural networks have recently shown strong performance for activity-coefficient modeling. However, extending such approaches to equilibrium data arising from an extremum principle, such as liquid-liquid equilibria, remains difficult. Here we present DISCOMAX, a differentiable algorithm for phase-equilibrium calculation that guarantees thermodynamic consistency at both training and inference, only subject to a user-specified discretization. The method combines discrete enumeration of feasible phase states with masked softmax aggregation in the backward pass, with the propagation of the true equilibrium state in the forward pass, using a straight-through gradient estimator to enable physics-consistent end-to-end learning of neural \gls{gE}-models. We show that this approach bears analogy to statistical thermodynamics, and we evaluate it on binary liquid-liquid equilibrium data where it outperforms existing surrogate-based methods, while offering a general framework for learning from different kinds of equilibrium data.

Figures & tables

\mathbf{x}^{\mathrm{aug}}=(x^{(i)})_{i=1}^{N^{\prime}}=(\mathbf{x},z),\quad N^{\prime}=N+1.

Algorithm 1 DISCOMAX: Differentiable Binary Equilibrium Solver with DISCRETE enumeration (fp) and masked SOFTMAX aggregation over feasible states (bp).

Model	MAE (mean $\pm$ std)	RMSE	$R^{2}$
DISCOMAX H+G	0.016 $\pm$ 0.017	0.018	0.998
DISCOMAX H	0.017 $\pm$ 0.018	0.019	0.998
DISCOMAX	0.019 $\pm$ 0.036	0.030	0.996
DISCOMAX G	0.031 $\pm$ 0.057	0.046	0.990
Surrogate H	0.109 $\pm$ 0.064	0.102	0.952
Surrogate H+G	0.109 $\pm$ 0.071	0.108	0.946

Table 1: Metrics of fitting 50 systems with different gap widths one by one (batch size = 1). H stands for the auxiliary Hessian loss, and G stands for the auxiliary Gibbs loss.

Model	Split	MAE	RMSE	$R^{2}$
DISCOMAX	Test	0.075 $\pm$ 0.003	0.098 $\pm$ 0.005	0.965 $\pm$ 0.003
	Validation	0.075 $\pm$ 0.003	0.098 $\pm$ 0.004	0.965 $\pm$ 0.003
	Train	0.035 $\pm<$ 0.001	0.040 $\pm<$ 0.001	0.993 $\pm<$ 0.001
DISCOMAX G	Test	0.070 $\pm$ 0.002	0.091 $\pm$ 0.005	0.969 $\pm$ 0.003
	Validation	0.070 $\pm$ 0.003	0.091 $\pm$ 0.005	0.969 $\pm$ 0.003
	Train	0.024 $\pm<$ 0.001	0.032 $\pm$ 0.001	0.995 $\pm<$ 0.001

Table 2: Full cross-validation results for the phase-composition predictions in the generalization study, aggregated over ten folds. Metrics are shown as mean

\pm

standard deviation.

Panel of Figure 5 — Figure 5: Test-set parity plots of phase compositions for representative model variants: a. DISCOMAX H, b. Surrogate G, and c. DISCOMAX. Full cross-validation results for all other variants are reported in Table 2 .

Panel of Figure 6 — Figure 6: Test-set gap-width parity plots for DISCOMAX H and Surrogate G on a representative fold.

Panel of Figure 7 — Figure 7: Training-set gap-width parity plots for DISCOMAX H and Surrogate G on the same fold as Figure 6 .

	Test MAE	Test RMSE	Test $R^{2}$
Model
DISCOMAX H	0.046 $\pm$ 0.003	0.047 $\pm$ 0.003	0.986 $\pm$ 0.002
DISCOMAX	0.045 $\pm$ 0.003	0.047 $\pm$ 0.003	0.986 $\pm$ 0.002

Table S1: Full cross-validation results for the test, validation, and train set aggregated over ten folds for the best performing models on the COSMO-RS dataset DISCOMAX batch-size 128 and DISCOMAX H batch-size 128. Metrics are shown as mean

\pm

standard deviation.

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Machine learning (ML) approaches have shown promising results for predicting molecular properties relevant for chemical process design. However, they are often limited by scarce experimental property data and lack thermodynamic consistency. As such, thermodynamics-informed ML, i.e., incorporating thermodynamic relations into the loss function as regularization term for training, has been proposed. We herein transfer the concept of thermodynamics-informed graph neural networks (GNNs) from the Gibbs-Duhem to the Clapeyron equation, predicting several pure component properties in a multi-task manner, namely: vapor pressure, liquid molar volume, vapor molar volume and enthalpy of vaporization. We find improved prediction accuracy of the Clapeyron-GNN compared to the single-task learning setting, and improved approximation of the Clapeyron equation compared to the purely data-driven multi-task learning setting. In fact, we observe the largest improvement in prediction accuracy for the properties with the lowest availability of data, making our model promising for practical application in data scarce scenarios of chemical engineering practice.

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