Learning the Infinitesimal Generator of Stochastic Diffusion Processes (2405.12940v1)

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.

• The paper introduces a novel energy-based framework in RKHS for learning the infinitesimal generator of SDEs, enhancing spectral estimation of system dynamics.
• It proposes a reduced-rank estimator with dimension-independent learning bounds, improving efficiency in high-dimensional settings.
• Empirical experiments on varied potentials demonstrate the method’s ability to accurately capture metastable states and outperform traditional transfer operators.

Learning Infinitesimal Generators of Stochastic Diffusion Processes

Introduction

When modeling real-world systems, we often use stochastic differential equations (SDEs) because they account for randomness in ways that traditional ordinary differential equations (ODEs) can't. This randomness is crucial in areas like finance, where SDEs model asset prices, and physics, where they describe atomic motion affected by thermal fluctuations.

The focus here is on learning the infinitesimal generator (IG) of these SDEs. The IG offers a detailed understanding of the system's dynamics and behavior that goes beyond identifying the usual drift and diffusion coefficients. Spectral decomposition of the IG can reveal important physical properties such as metastable states and time scale separations.

Key Contributions

The paper makes several key contributions:

1. Novel Framework: It introduces a framework for learning the IG in reproducing kernel Hilbert spaces (RKHS) using an energy-based risk metric.
2. Reduced-Rank Estimator: It proposes a reduced-rank estimator with learning bounds independent of state space dimension.
3. Spectral Learning Bounds: Establishes new spectral learning bounds for IG estimation.
4. Practical Advantages: Demonstrates the practical superiority of this approach compared to previous methods.

Background

For continuous-time processes, SDEs are often expressed as: $dX_t = a(X_{t})dt + b(X_{t})dW_t,$ where $a$ and $b$ are the drift and diffusion coefficients, respectively. Learning these coefficients from observed data isn't new, but this paper shifts the focus towards learning the IG, which provides a richer understanding of system dynamics.

The IG can theoretically be learned using Transfer Operators (TO), which describe the average evolution of state functions over time. However, TO methods require evenly sampled data and are purely data-driven, making them impractical in some scenarios. This spurred interest in directly learning the IG, which can handle uneven sampling and incorporate knowledge of the SDE.

Novel Statistical Learning Framework

The primary challenge with IGs is that they are unbounded operators, unlike the well-behaved Hilbert-Schmidt operators used for TOs. The paper overcomes this by introducing a risk metric based on the system's energy function. Here’s a streamlined process of their framework:

1. Energy-Based Risk: Define a risk functional that can be minimized to produce good spectral estimates. This involves embedding the IG in an RKHS and using known physical properties to guide the learning process.
2. Resolvent Operator: Instead of learning the IG directly, learn its compact resolvent, which shares the same eigenfunctions and can be approximated by finite-rank operators.
3. Empirical Estimators: Develop empirical risk minimization (ERM) strategies, using regularized operators that handle both full and partial knowledge settings.

Empirical Risk Minimization

The authors provide two main estimators:

1. Kernel Ridge Regression (KRR): Suited for situations where the full rank is unconstrained.
2. Reduced Rank Regression (RRR): Adds a rank constraint to the learning process, making it more efficient for high-dimensional data.

By utilizing empirical estimates of covariance operators and adopting regularization techniques, the paper presents a mathematically rigorous and practically effective method for learning IGs from data.

Spectral Learning Bounds

An exciting aspect of this work is the presentation of spectral learning bounds, which ensure the stability and accuracy of the learned IG. These bounds are crucial for practical applications as they guarantee that the learnt operator will provide reliable estimates of eigenvalues and eigenfunctions.

Experiments

The paper validates its methods with three key experiments:

1. One Dimensional Four Well Potential: Demonstrates the absence of spurious eigenvalues, showing the robustness of the RRR method compared to other approaches.
2. Muller Brown Potential: Highlights the method's ability to recover metastable states better than Transfer Operators.
3. Cox-Ingersoll-Ross (CIR) Model: Validates the approach for financial models and showcases the effective prediction of conditional expectations.

These experiments illustrate the practical efficiency and accuracy of the proposed framework.

Implications and Future Directions

This work has both theoretical and practical implications.

• Theoretically: It offers a novel approach to learning unbounded operators, extending the toolkit available to researchers working with complex dynamical systems.
• Practically: This method can substantially improve modeling in fields like finance and physics, where understanding the intricate dynamics of a system is critical.

Conclusion

This paper provides a robust, theoretically grounded method for learning the IG of SDEs. By addressing the challenges posed by the unbounded nature of IGs, it opens new avenues for accurately modeling and understanding complex stochastic systems. Future work could explore further optimization of the framework and its application to more diverse types of stochastic processes.