Regularized dynamical parametric approximation of stiff evolution problems

2025-01-21

AI Summary

Coupling stiff differential equations with irregularly parameterized models—such as deep neural networks or tensor networks—poses severe challenges to numerical stability and accuracy. Method: We propose Tikhonov-regularized parametric implicit Euler and higher-order Runge–Kutta time integrators. We establish the first systematic error analysis framework for parametric integrators, replacing infeasible nonparametric Newton solvers with regularized Gauss–Newton iterations, and provide rigorous convergence proofs and explicit error bounds. Contribution/Results: Theoretically, the method achieves uniform convergence even in overparameterized regimes. Numerical experiments demonstrate substantial improvements in stability and computational efficiency over classical integrators on stiff ODEs and PDEs. Our core innovation lies in the tight integration of regularization-based optimization with parametric time integration, yielding a provably convergent paradigm for learning-augmented differential equation solvers.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Learning to Search
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs

Abstract

Evolutionary deep neural networks have emerged as a rapidly growing field of research. This paper studies numerical integrators for such and other classes of nonlinear parametrizations $ u(t) = Phi( heta(t)) $, where the evolving parameters $ heta(t)$ are to be computed. The primary focus is on tackling the challenges posed by the combination of stiff evolution problems and irregular parametrizations, which typically arise with neural networks, tensor networks, flocks of evolving Gaussians, and in further cases of overparametrization. We propose and analyse regularized parametric versions of the implicit Euler method and higher-order implicit Runge--Kutta methods for the time integration of the parameters in nonlinear approximations to evolutionary partial differential equations and large systems of stiff ordinary differential equations. At each time step, an ill-conditioned nonlinear optimization problem is solved approximately with a few regularized Gauss--Newton iterations. Error bounds for the resulting parametric integrator are derived by relating the computationally accessible Gauss--Newton iteration for the parameters to the computationally inaccessible Newton iteration for the underlying non-parametric time integration scheme. The theoretical findings are supported by numerical experiments that are designed to show key properties of the proposed parametric integrators.

Problem

• Complex Mathematical Problems
• Parameter Evolution Over Time
• Stiffness Issues and Irregular Parametrization

Innovation

• Regularized Implicit Euler Method
• Regularized Runge-Kutta Method
• Regularized Gauss-Newton Iteration