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This research addresses parameter non-uniqueness in redundant models like neural networks by introducing inertia into Dirac-Frenkel dynamics for evolution problems. The authors prove the inertial formulation yields well-posed parameter dynamics and demonstrate improved robustness through numerical experiments, while maintaining compatibility with standard regularized linear least-squares solvers.
Dirac-Frenkel dynamics with inertia for nonlinearly parametrized solutions of evolution problems
arXiv:2606.24769v2 Announce Type: replace-cross 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.
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.
Numerical Analysis (math.NA); Machine Learning (cs.LG)
arXiv:2606.24769 [math.NA]
(or arXiv:2606.24769v2 [math.NA] for this version)
https://doi.org/10.48550/arXiv.2606.24769 Focus to learn more arXiv-issued DOI via DataCite
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