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Variational estimation of generator invariant subspaces
Luca Donati1,2, Fazil Safarov1,2, Surahit Chewle2
1Freie Universität Berlin, Department of Mathematics and Computer Science, Arnimallee 22, D-14195 Berlin, Germany.
The Journal of Chemical Physics
|July 23, 2026
Summary
We introduce Variational Estimation of Generator Invariant Subspaces (VEGIS), a new method using neural networks to find key dynamic modes in complex systems. This approach aids in understanding molecular processes and improving simulation sampling strategies.
Area of Science:
- Computational chemistry
- Statistical mechanics
- Machine learning
Background:
- Diffusion processes are fundamental to molecular dynamics.
- Approximating invariant subspaces of the generator is crucial for analyzing slow dynamics.
- Current methods often struggle with high-dimensional systems.
Purpose of the Study:
- To develop a variational method for approximating invariant subspaces of diffusion process generators.
- To enable the recovery of eigenfunctions, eigenvalues, and metastable sets.
- To introduce an enhanced sampling strategy based on identified slow modes.
Main Methods:
- Utilizing neural networks to represent trial subspaces.
- Optimizing a Dirichlet-form trace objective with equilibrium samples and network gradients.
- Employing diagonalization and PCCA+ for analysis.
- Implementing a VEGIS-driven sampling strategy.
Main Results:
- VEGIS accurately approximates invariant subspaces.
- The method recovers generator eigenfunctions and eigenvalues.
- It successfully identifies metastable sets via membership functions.
- The VEGIS-driven sampling strategy improves efficiency.
Conclusions:
- VEGIS offers an accurate and flexible approach to analyzing complex dynamical systems.
- The method is applicable to both low-dimensional and molecular systems.
- VEGIS enhances our ability to understand and simulate molecular dynamics.
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