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arXiv · 2606.24769

Dirac-Frenkel dynamics with inertia for nonlinearly parametrized solutions of evolution problems

Matteo Raviola·Benjamin Peherstorfer
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Abstract

Even when Dirac-Frenkel dynamics determine a well-defined evolution in function space, the corresponding parameter dynamics can be non-unique or ill-conditioned for redundant nonlinear parametrizations, such as typical neural networks or mixture models. We propose to add inertia to the Dirac-Frenkel dynamics and show that this allows useful parameter velocity information to persist from the past trajectory in directions that are weakly informed, while well-informed parameter velocity directions continue to follow the Dirac-Frenkel dynamics. We prove that the inertial formulation yields well-posed parameter dynamics and provide a posteriori error bounds. After time discretization, the method requires the solution of the same type of regularized linear least-squares problem as standard Dirac-Frenkel dynamics, but with the previous velocity appearing as an anchor. Numerical experiments demonstrate the increased robustness obtained with inertia.

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Matteo Raviola, Benjamin Peherstorfer. 2026-08-07. Dirac-Frenkel dynamics with inertia for nonlinearly parametrized solutions of evolution problems. https://arxiv.org/abs/2606.24769

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