A Dirac-Frenkel-Onsager principle: Instantaneous residual minimization with gauge momentum for nonlinear parametrizations of PDE solutions

2026-05-04 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Scientific Machine Learning · Depth: Expert, extended

Summary

The Dirac-Frenkel (DF) principle, used for evolving nonlinear parametrizations of PDE solutions, often faces challenges like non-unique parameter dynamics and ill-conditioning, particularly during "tangent space collapse." This paper introduces the Dirac-Frenkel-Onsager (DFO) principle, which addresses these issues by interpreting the non-uniqueness as "gauge freedom" and leveraging Onsager's minimum-dissipation principle to inject a history variable, or "momentum," exclusively along the nullspace directions of the parametrization Jacobian. This approach ensures instantaneous residual minimization, avoiding the bias introduced by standard regularization, while promoting temporally smooth parameter evolutions and enhancing robustness in singular regimes. Numerical experiments across various low- and high-dimensional PDEs, including wave collision and Fokker-Planck equations, demonstrate that DFO achieves significantly lower errors, smoother parameter trajectories, and reduced background noise with negligible additional computational cost compared to existing DF-based methods.

Key takeaway

The Dirac-Frenkel-Onsager (DFO) principle resolves non-unique and ill-conditioned parameter dynamics in neural PDE solvers by injecting Onsager-type momentum only along the Jacobian's nullspace directions. This approach preserves instantaneous residual minimization, unlike standard regularization, and yields orders of magnitude lower errors and smoother parameter evolutions in singular regimes. DFO offers a robust, low-cost solution for accurate and stable time-evolution of nonlinear PDE parametrizations across various scientific and engineering applications.

Topics

• Dirac-Frenkel-Onsager Principle
• Dirac-Frenkel Variational Principle
• Nonlinear PDE Parametrizations
• Gauge Freedom
• Tangent Space Collapse

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.