A systematic comparison of hard- and soft-constrained physics-informed molecular machine learning with the Clapeyron equation
Abstract
In molecular machine learning (ML), significant progress has been made on integrating fundamental thermodynamic relations into ML models. Such physics-informed approaches can be categorized as hard-constrained, where the thermodynamic relations are embedded in the model architecture, and soft-constrained, where relations are included in the training loss. While both approaches have been applied to various property prediction tasks, a systematic comparison on experimental data is lacking. We herein compare hard- and soft-constrained approaches on a numerically challenging case study: the prediction of vapor pressure, saturated liquid and vapor molar volumes, and enthalpy of vaporization as functions of temperature for single-species vapor-liquid equilibrium, related by the exact Clapeyron equation. Furthermore, soft-constrained approaches in molecular ML typically balance data and physics losses with a fixed weighting factor, requiring careful tuning. To overcome this, we investigate the augmented Lagrangian method (ALM), well established in constrained optimization. We find that for the soft-constrained approach, using the ALM improves prediction performance in terms of constraint satisfaction by factor 2 and reduces training epochs compared to using a fixed penalty by 30%. Hence, in the soft-constrained approach, the use of the ALM is always recommended. The hard-constrained approach, which entails additional architectural design choices, can reach on par accuracy with the soft-constrained approach, at even higher thermodynamic consistency, reaching 12 orders of magnitude closer approximation of the Clapeyron equation. Overall, the hard-constrained approach is most promising when thermodynamic consistency is critical, yet requires tailoring the ML model architecture to the respective thermodynamic relation.
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BibTeX
@misc{pavsek2026systematic,
title = {{A systematic comparison of hard- and soft-constrained physics-informed molecular machine learning with the Clapeyron equation}},
author = {Jan Pavšek and Jan G. Rittig and Alexander Mitsos},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.36947},
eprint = {2609.36947},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.36947},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A systematic comparison of hard- and soft-constrained physics-informed molecular machine learning with the Clapeyron equation
AU - Jan Pavšek
AU - Jan G. Rittig
AU - Alexander Mitsos
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.36947
UR - https://arxiv.org/abs/2609.36947
AB - In molecular machine learning (ML), significant progress has been made on integrating fundamental thermodynamic relations into ML models. Such physics-informed approaches can be categorized as hard-constrained, where the thermodynamic relations are embedded in the model architecture, and soft-constrained, where relations are included in the training loss. While both approaches have been applied to various property prediction tasks, a systematic comparison on experimental data is lacking. We herein compare hard- and soft-constrained approaches on a numerically challenging case study: the prediction of vapor pressure, saturated liquid and vapor molar volumes, and enthalpy of vaporization as functions of temperature for single-species vapor-liquid equilibrium, related by the exact Clapeyron equation. Furthermore, soft-constrained approaches in molecular ML typically balance data and physics losses with a fixed weighting factor, requiring careful tuning. To overcome this, we investigate the augmented Lagrangian method (ALM), well established in constrained optimization. We find that for the soft-constrained approach, using the ALM improves prediction performance in terms of constraint satisfaction by factor 2 and reduces training epochs compared to using a fixed penalty by 30%. Hence, in the soft-constrained approach, the use of the ALM is always recommended. The hard-constrained approach, which entails additional architectural design choices, can reach on par accuracy with the soft-constrained approach, at even higher thermodynamic consistency, reaching 12 orders of magnitude closer approximation of the Clapeyron equation. Overall, the hard-constrained approach is most promising when thermodynamic consistency is critical, yet requires tailoring the ML model architecture to the respective thermodynamic relation.
ER -