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Differentiable learning and control of free-energy-driven pattern dynamics

Alexander E. Cohen1,*, Samuel Degnan-Morgenstern1,†, Simon Daubner2, Jörn Dunkel3,‡, and Martin Z. Bazant3,1,§

  • *Contact author: aecohen@mit.edu
  • †Contact author: stm16109@mit.edu
  • ‡Contact author: dunkel@mit.edu
  • §Contact author: bazant@mit.edu
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Phys. Rev. Research 8, 023344 – Published 25 June, 2026

DOI: https://doi.org/10.1103/b8kc-vpwq

Abstract

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.

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Phys. Rev. Research 8, 023344– Published 25 June, 2026
  • Received 20 March 2026
  • Accepted 19 May 2026
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