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Herglotz Lagrangian neural networks: Learning dissipative dynamics via contact variational principles

Qi Huang1, Jack Murdoch Moore1,2, Ting-Ting Gao3,*, and Gang Yan1,2,†

  • 1MOE Key Laboratory of Advanced Micro-Structured Materials, and School of Physical Science and Engineering, Tongji University, Shanghai 200092, People's Republic of China
  • 2National Key Laboratory of Autonomous Intelligent Unmanned Systems, MOE Frontiers Science Center for Intelligent Autonomous Systems, Tongji University, Shanghai 200092, People’s Republic of China
  • 3Network Science Institute, Northeastern University, Boston, Massachusetts 02115, USA

  • *Contact author: ti.gao@northeastern.edu
  • †Contact author: gyan@tongji.edu.cn
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Phys. Rev. Research 8, 023224 – Published 29 May, 2026

DOI: https://doi.org/10.1103/9gnh-89jd

Abstract

Physics-informed machine learning has proven effective for learning conservative dynamical systems. However, most real-world systems are nonconservative, and current approaches typically handle dissipation through ad hoc modifications or require access to full-state information. Here, we introduce the discrete Herglotz Lagrangian neural network (HLNN), a structure-preserving framework for learning a class of smooth dissipative dynamics from data. Grounded in the Herglotz variational principle, HLNN extends classical variational learning to dissipative systems by modeling dissipation intrinsically within a contact-geometric framework, without prescribing an explicit empirical dissipation law, and can be trained from position-only observations without explicit velocity supervision. Across benchmarks spanning state-dependent damping, time-modulated damping, and external driving, HLNN reliably recovers long-term trajectories and dissipation rates, outperforming baseline models on the considered benchmarks. Our results establish a unified variational and geometric framework for learning smooth dissipative dynamics and open avenues for data-driven modeling of nonequilibrium and open systems.

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Phys. Rev. Research 8, 023224– Published 29 May, 2026
  • Received 12 January 2026
  • Accepted 10 May 2026
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