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Learning the Infinitesimal Generator of Stochastic Diffusion Processes
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Proposes an energy-based risk and a reduced-rank RKHS estimator for learning the infinitesimal generator of stochastic diffusions, with dimension-free spectral learning bounds and no spurious eigenvalues.
Problem
Learning the spectrum (eigenvalues and eigenfunctions) of the unbounded infinitesimal generator of stochastic diffusion processes from data, where classical Hilbert-Schmidt analysis fails and existing estimators produce spurious eigenvalues without learning guarantees.
Approach
Replace the standard L2 risk with an energy-based risk measured in a shifted Sobolev-type energy space W^mu, estimable from samples when the Dirichlet form (diffusion and Dirichlet operator) is known; learn the resolvent with a reduced-rank regression estimator in an RKHS and transfer errors to eigenpairs via perturbation theory.
Key results
- On a four-well Langevin potential, the RRR estimator shows no spurious eigenvalues, unlike the KRR generator estimator.
- On Muller-Brown Langevin dynamics, the method recovers the second eigenfunction where the transfer operator approach fails to recover the metastable state.
- For the CIR model the paper reports prediction RMSE decreasing with number of samples, used to validate the learning bounds.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use RRR instead of KRR | Generator resolvent estimation · Four-well Langevin | spurious eigenvalue estimation | Helped |
| use physics-informed RRR instead of transfer-operator learning | Second eigenfunction estimation · Muller Brown potential | eigenfunction recovery | Helped |
| use transfer-operator learning instead of physics-informed RRR | Second eigenfunction estimation · Muller Brown potential | metastable-state recovery | Hurt |
| choose hyperparameters with higher empirical bias | RRR · Four-well Langevin | operator estimation quality | Hurt |
Limitations
- Computational complexity scales as n^2 d^2, limiting scalability
- Requires known Dirichlet form (diffusion and Dirichlet operator) for partial knowledge setting
- Boundedness assumption (BK) on the Dirichlet embedding is the main limiting factor; regularity needs alpha>=1 for spectral bounds
Experimental setup
Datasets: simulated Langevin trajectories (four-well potential) simulated Muller-Brown potential Langevin data simulated CIR process data
Benchmarks: four-well Langevin eigenvalue/eigenfunction estimation Muller-Brown second eigenfunction CIR conditional expectation RMSE
Metrics: eigenvalue estimation error eigenfunction accuracy RMSE
Baselines: kernel Galerkin / KRR generator estimator transfer operator (TO) method extended dynamical mode decomposition
Builds on: Galerkin method for generator kernel transfer operator regression kernel ridge regression of generator (Kostic et al.) spectral perturbation theory
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