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Learning the Infinitesimal Generator of Stochastic Diffusion Processes

Vladimir R. Kostic, Karim Lounici, Helene Halconruy, Timothee Devergne, Massimiliano Pontil

TL;DR

The paper tackles the problem of data-driven learning of the infinitesimal generator $L$ for stochastic diffusion processes, addressing the unboundedness of $L$ by reframing learning around the resolvent $(\mu I - L)^{-1}$ within an energy space. It introduces an energy-based risk in RKHS, leverages the embedding $Z_\\mu$ into the energy space $\\mathcal{W}^{\\mu}_\\pi(\\mathcal{X})$, and derives two estimators, KRR and RRR, to learn a finite-rank approximation of the resolvent. Theoretical contributions include the first spectral learning bounds for generator learning, decomposition of errors into regularization, rank-reduction, and variance terms with rates depending on regularity and embedding properties, and minimax-like guarantees under suitable conditions. Empirically, the approach yields accurate eigenpairs for diffusion generators, avoids spurious eigenvalues, and outperforms transfer-operator-based methods on challenging examples such as Langevin dynamics and CIR, while highlighting scalability challenges and directions for broader SDEs.

Abstract

We address data-driven learning of the infinitesimal generator of stochastic diffusion processes, essential for understanding numerical simulations of natural and physical systems. The unbounded nature of the generator poses significant challenges, rendering conventional analysis techniques for Hilbert-Schmidt operators ineffective. To overcome this, we introduce a novel framework based on the energy functional for these stochastic processes. Our approach integrates physical priors through an energy-based risk metric in both full and partial knowledge settings. We evaluate the statistical performance of a reduced-rank estimator in reproducing kernel Hilbert spaces (RKHS) in the partial knowledge setting. Notably, our approach provides learning bounds independent of the state space dimension and ensures non-spurious spectral estimation. Additionally, we elucidate how the distortion between the intrinsic energy-induced metric of the stochastic diffusion and the RKHS metric used for generator estimation impacts the spectral learning bounds.

Learning the Infinitesimal Generator of Stochastic Diffusion Processes

TL;DR

The paper tackles the problem of data-driven learning of the infinitesimal generator for stochastic diffusion processes, addressing the unboundedness of by reframing learning around the resolvent within an energy space. It introduces an energy-based risk in RKHS, leverages the embedding into the energy space , and derives two estimators, KRR and RRR, to learn a finite-rank approximation of the resolvent. Theoretical contributions include the first spectral learning bounds for generator learning, decomposition of errors into regularization, rank-reduction, and variance terms with rates depending on regularity and embedding properties, and minimax-like guarantees under suitable conditions. Empirically, the approach yields accurate eigenpairs for diffusion generators, avoids spurious eigenvalues, and outperforms transfer-operator-based methods on challenging examples such as Langevin dynamics and CIR, while highlighting scalability challenges and directions for broader SDEs.

Abstract

We address data-driven learning of the infinitesimal generator of stochastic diffusion processes, essential for understanding numerical simulations of natural and physical systems. The unbounded nature of the generator poses significant challenges, rendering conventional analysis techniques for Hilbert-Schmidt operators ineffective. To overcome this, we introduce a novel framework based on the energy functional for these stochastic processes. Our approach integrates physical priors through an energy-based risk metric in both full and partial knowledge settings. We evaluate the statistical performance of a reduced-rank estimator in reproducing kernel Hilbert spaces (RKHS) in the partial knowledge setting. Notably, our approach provides learning bounds independent of the state space dimension and ensures non-spurious spectral estimation. Additionally, we elucidate how the distortion between the intrinsic energy-induced metric of the stochastic diffusion and the RKHS metric used for generator estimation impacts the spectral learning bounds.
Paper Structure (28 sections, 27 theorems, 161 equations, 3 figures, 1 table)

This paper contains 28 sections, 27 theorems, 161 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background and the problem
  3. Novel statistical learning framework
  4. Empirical risk minimization
  5. Spectral learning bounds
  6. Experiments
  7. Conclusion
  8. Background
  9. Basics on operator theory for Markov processes.
  10. Infinitesimal generator for diffusion processes
  11. Spectral perturbation theory
  12. Representations in the RKHS
  13. Statistical learning framework
  14. Spectral perturbation bounds
  15. Empirical estimation
  16. ...and 13 more sections

Key Result

Proposition 1

Given $\mu>0$, let $\mathcal{H}\!\subseteq\!\mathcal{W}^{\mu}_\pi(\mathcal{X})$ be the RKHS associated to kernel $k\in\mathcal{C}^{2}(\mathcal{X}\!\times\!\mathcal{X})$ such that $Z_\mu\in{\rm{HS}}\left(\mathcal{H},\mathcal{W}^{\mu}_\pi(\mathcal{X})\right)$, and let $P_\mathcal{H}$ be the orthogonal

Figures (3)

  • Figure 1: a) Empirical biases $\hat{s}_1 = \widehat{\sigma}_1\,\widehat{\eta}_1$ and estimation of the first (nontrivial) eigenfunction of the IG of a Langevin process under a four well potential. Ground truth is black, our method RRR is red and blue for two different kernel lengthscales. b) Eigenvalue estimation for the same process compared to the methods in hou23cCabannes_2023_Galerkin, for which eigenvalue histogram in blue shows spuriousness. c) Estimation of the second eigenfunction of a Langevin process under Muller brown potential (white level lines) by RRR, Transfer Operator (TO) in d) and ground truth in e). Observe that TO fails to recover the metastable state. f) Prediction RMSE for the CIR model.
  • Figure 2: Results of the RRR given by our method for two different length scales (blue and red) compared with ground truth (black) for the Langevin dynamics driven by a four well one dimensional potential.
  • Figure 3: Results of the RRR given by our method for two different length scales (blue and red) compared with ground truth for the Langevin dynamics driven by a four well one dimensional potential.

Theorems & Definitions (52)

  • Example 1: Langevin
  • Example 2: Cox-Ingersoll-Ross process
  • Proposition 1
  • Proposition 2
  • Theorem 1
  • Theorem 2
  • Remark 1
  • Example 3: Langevin
  • Example 4: Cox-Ingersoll-Ross process
  • Proposition 3: DK1970
  • ...and 42 more