@article{Cohen:2026:10.1103/b8kc-vpwq,
author = {Cohen, AE and Degnan-Morgenstern, S and Daubner, S and Dunkel, J and Bazant, MZ},
doi = {10.1103/b8kc-vpwq},
journal = {Physical Review Research},
title = {Differentiable learning and control of free-energy-driven pattern dynamics},
url = {http://dx.doi.org/10.1103/b8kc-vpwq},
volume = {8},
year = {2026}
}

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TY  - JOUR
AB  - <jats:p>Pattern-forming dynamics govern the behavior of many quantum, electrochemical, and soft-matter systems, yet learning and controlling the corresponding partial differential equation (PDE) models on realistic geometries remains challenging. In this work, we develop a unified, differentiable framework for free-energy-based PDEs that enables end-to-end parameter inference and optimal control directly on image-based domains. Starting from a variational description in terms of an energy or free energy functional, we combine PDE-based smoothing of segmented geometries, the smoothed boundary method, advanced integrators for stiff pattern-forming PDEs, and memory-efficient automatic differentiation implemented in JAX to construct scalable PDE solvers amenable to gradient-based optimization. We demonstrate the capabilities of this framework across four classes of applications: (1) learning Cahn-Hilliard and Allen-Cahn free energies and kinetic laws from noisy spatiotemporal data on complex battery electrode microstructures; (2) designing time-dependent wetting boundary conditions that steer phase boundaries to desired orientations; (3) optimizing spatially varying reaction rates, interpreted as surface coatings, to suppress phase separation in intercalation electrode models; and (4) computing time-dependent trapping potentials that transfer Bose-Einstein condensates between ground states of qualitatively different Gross-Pitaevskii potentials while minimizing excitations. Together, these results show that variational PDE models, when equipped with differentiable solvers on complex domains, provide a versatile substrate for data-driven discovery, design, and control of pattern-forming materials and quantum fluids, and they point toward tighter integration of physics-based PDE modeling with modern optimization and machine learning pipelines.</jats:p>
AU  - Cohen,AE
AU  - Degnan-Morgenstern,S
AU  - Daubner,S
AU  - Dunkel,J
AU  - Bazant,MZ
DO  - 10.1103/b8kc-vpwq
PY  - 2026///
TI  - Differentiable learning and control of free-energy-driven pattern dynamics
T2  - Physical Review Research
UR  - http://dx.doi.org/10.1103/b8kc-vpwq
UR  - https://doi.org/10.1103/b8kc-vpwq
VL  - 8
ER  - 

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