TIP-INN: A thermodynamics-informed machine learning framework for the mechanical behavior of rocks
6 affiliations
- 1Department of Geosciences, The Pennsylvania State University , University Park, Pennsylvania, 16802 ,
- 2Lamont Doherty Earth Observatory, Columbia University , Palisades, New York, 10964 ,
- 3Earth Resources Laboratory, Massachusetts Institute of Technology , Cambridge, MA, 02139 ,
- 4Department of Earth and Atmospheric Sciences, Cornell University , Ithaca, New York, 14850 ,
- 5GFZ Helmholtz Center for Geosciences , Telegrafenberg, 14473 Potsdam ,
- 6Laboratoire de Mécanique et Génie Civil, Université de Montpellier ,
Abstract
Modeling the mechanical deformation of geological materials across relevant thermodynamic conditions and dynamic processes remains a fundamental challenge in Earth sciences. Conventional rheological models rely on isolated disciplinary approaches and hard-coded iterative solvers that struggle with strongly coupled, non-linear constitutive laws. Inverting macroscopic laboratory data to infer sub-grid micro-physics is also ill-posed, producing structural non-uniqueness and models that may violate thermodynamic laws during transient loading. To address these limitations, we present TIP-INN (Thermodynamics of Irreversible Processes Informed Neural Network), a thermodynamically consistent machine learning framework grounded in the Generalized Standard Materials (GSM) formalism. TIP-INN discovers constitutive laws directly from time-series stress-strain data by learning two strictly convex thermodynamic potentials-the Helmholtz free energy and the dual dissipation potential-architecturally guaranteeing non-negative mechanical dissipation. A continuous, differentiable neural integrator solves internal state variable (ISV) evolution as an initial-value problem, enabling stable, rapid training across complex rheologies. Benchmarked against existing thermodynamics-informed networks, TIP-INN trains 10 to 30 times faster than variational baselines and reduces extrapolation error by up to two orders of magnitude. We resolve the non-uniqueness of this inverse problem through two strategies: (1) diverse loading protocols (e.g., cyclic and stress-relaxation paths) that map the full geometry of the potential state space, and (2) a multi-modal loss assimilating auxiliary seismo-acoustic data. TIP-INN thus rigorously bridges continuum damage mechanics and laboratory seismology. Finally, to ensure physical interpretability, the learned potentials are distilled via Sparse Identification of Nonlinear Potentials (SINP) into closed-form expressions for efficient implementation in geodynamic codes.
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BibTeX
@article{li2026tipinn,
title = {{TIP-INN: A thermodynamics-informed machine learning framework for the mechanical behavior of rocks}},
author = {Kangan Li and Tushar Mittal and Benjamin Holtzman and Seth Saltiel and Raphael Affinito and Loic Daridon},
journal = {PNAS Nexus},
year = {2026},
month = oct,
doi = {10.1093/pnasnexus/pgag349},
url = {https://academic.oup.com/pnasnexus/advance-article/doi/10.1093/pnasnexus/pgag349/8898566},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - TIP-INN: A thermodynamics-informed machine learning framework for the mechanical behavior of rocks
AU - Kangan Li
AU - Tushar Mittal
AU - Benjamin Holtzman
AU - Seth Saltiel
AU - Raphael Affinito
AU - Loic Daridon
PY - 2026
DA - 2026/10/09
JO - PNAS Nexus
DO - 10.1093/pnasnexus/pgag349
UR - https://academic.oup.com/pnasnexus/advance-article/doi/10.1093/pnasnexus/pgag349/8898566
AB - Modeling the mechanical deformation of geological materials across relevant thermodynamic conditions and dynamic processes remains a fundamental challenge in Earth sciences. Conventional rheological models rely on isolated disciplinary approaches and hard-coded iterative solvers that struggle with strongly coupled, non-linear constitutive laws. Inverting macroscopic laboratory data to infer sub-grid micro-physics is also ill-posed, producing structural non-uniqueness and models that may violate thermodynamic laws during transient loading. To address these limitations, we present TIP-INN (Thermodynamics of Irreversible Processes Informed Neural Network), a thermodynamically consistent machine learning framework grounded in the Generalized Standard Materials (GSM) formalism. TIP-INN discovers constitutive laws directly from time-series stress-strain data by learning two strictly convex thermodynamic potentials-the Helmholtz free energy and the dual dissipation potential-architecturally guaranteeing non-negative mechanical dissipation. A continuous, differentiable neural integrator solves internal state variable (ISV) evolution as an initial-value problem, enabling stable, rapid training across complex rheologies. Benchmarked against existing thermodynamics-informed networks, TIP-INN trains 10 to 30 times faster than variational baselines and reduces extrapolation error by up to two orders of magnitude. We resolve the non-uniqueness of this inverse problem through two strategies: (1) diverse loading protocols (e.g., cyclic and stress-relaxation paths) that map the full geometry of the potential state space, and (2) a multi-modal loss assimilating auxiliary seismo-acoustic data. TIP-INN thus rigorously bridges continuum damage mechanics and laboratory seismology. Finally, to ensure physical interpretability, the learned potentials are distilled via Sparse Identification of Nonlinear Potentials (SINP) into closed-form expressions for efficient implementation in geodynamic codes.
SN - 2752-6542
ER -