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

arXiv: https://arxiv.org/abs/2606.24769 · HTML: https://mlsanity.com/paper/2606.24769/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 39 references that are in the map.
- First posted on arXiv 23 Jun 2026.

## Builds on (most cited first, 15 of 15)

- [Neural Galerkin Schemes with Active Learning for High-Dimensional Evolution Equations](https://mlsanity.com/paper/2203.01360/) (2022)
- [Evolutional Deep Neural Network](https://mlsanity.com/paper/2103.09959/) (2021)
- [Implicit Neural Spatial Representations for Time-dependent PDEs](https://mlsanity.com/paper/2210.00124/) (2022)
- [A Kaczmarz-inspired approach to accelerate the optimization of neural network wavefunctions](https://mlsanity.com/paper/2401.10190/) (2024)
- [Randomized Sparse Neural Galerkin Schemes for Solving Evolution Equations with Deep Networks](https://mlsanity.com/paper/2310.04867/) (2023)
- [A Stable and Scalable Method for Solving Initial Value PDEs with Neural Networks](https://mlsanity.com/paper/2304.14994/) (2023)
- [Particle-based Energetic Variational Inference](https://mlsanity.com/paper/2004.06443/) (2020)
- [Improving Energy Natural Gradient Descent through Woodbury, Momentum, and Randomization](https://mlsanity.com/paper/2505.12149/) (2025)
- [Effective Dimension Adaptive Sketching Methods for Faster Regularized Least-Squares Optimization](https://mlsanity.com/paper/2006.05874/) (2020)
- [Sequential-in-time training of nonlinear parametrizations for solving time-dependent partial differential equations](https://mlsanity.com/paper/2404.01145/) (2024)
- [Optimal Rates of Sketched-regularized Algorithms for Least-Squares Regression over Hilbert Spaces](https://mlsanity.com/paper/1803.04371/) (2018)
- [Regularized dynamical parametric approximation of stiff evolution problems](https://mlsanity.com/paper/2501.12118/) (2025)
- [Natural gradient descent with momentum](https://mlsanity.com/paper/2604.15554/) (2026)
- [Nonlinear model reduction for transport-dominated problems](https://mlsanity.com/paper/2602.01397/) (2026)
- [A Dirac-Frenkel-Onsager principle: Instantaneous residual minimization with gauge momentum for nonlinear parametrizations of PDE solutions](https://mlsanity.com/paper/2605.00284/) (2026)

## Abstract (arXiv)

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.