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SPINNs -- Deep learning framework for approximation of stochastic differential equations

Marcin Baranek, Paweł Przybyłowicz

TL;DR

This work extends physics-informed neural networks to stochastic differential equations driven by additive Lévy noise by formulating a pathwise, deterministic surrogate for the stochastic dynamics. It transforms the SDE into a random ODE using Y=X−σL and learns the map Ψ from Lévy trajectories to the solution with a neural network, guided by two loss formulations and Lévy-bridge based empirical losses optimized via SGD. Numerical experiments on Wiener-driven examples demonstrate that SPINNs can closely reproduce both trajectories and dynamics, with initial conditions enforced through architectural constraints. The framework lays groundwork for rigorous convergence analysis and points toward extensions to multiplicative noise via Doss–Sussman transformations.

Abstract

In this paper, we introduce the SPINNs (stochastic physics-informed neural networks) in a systematic manner. This provides a mathematical framework for approximating the solution of stochastic differential equations (SDEs) driven by Levy noise using artificial neural networks.

SPINNs -- Deep learning framework for approximation of stochastic differential equations

TL;DR

This work extends physics-informed neural networks to stochastic differential equations driven by additive Lévy noise by formulating a pathwise, deterministic surrogate for the stochastic dynamics. It transforms the SDE into a random ODE using Y=X−σL and learns the map Ψ from Lévy trajectories to the solution with a neural network, guided by two loss formulations and Lévy-bridge based empirical losses optimized via SGD. Numerical experiments on Wiener-driven examples demonstrate that SPINNs can closely reproduce both trajectories and dynamics, with initial conditions enforced through architectural constraints. The framework lays groundwork for rigorous convergence analysis and points toward extensions to multiplicative noise via Doss–Sussman transformations.

Abstract

In this paper, we introduce the SPINNs (stochastic physics-informed neural networks) in a systematic manner. This provides a mathematical framework for approximating the solution of stochastic differential equations (SDEs) driven by Levy noise using artificial neural networks.

Paper Structure

This paper contains 13 sections, 6 theorems, 54 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Transforming SDE to RODE
  4. Reformulation of solving SDE as a stochastic optimization problem
  5. Two loss functions
  6. SGD algorithm
  7. Some insight into the multiplicative noise case
  8. Numerical experiments
  9. Implementation details
  10. training details.
  11. results
  12. Appendix
  13. Conclusion and future work

Key Result

Proposition 1

Under assumption (A1) the following holds for all $y\in \mathcal{C}^1\left([0,T]\times\Omega;\mathbb{R}^d\right)$:

Figures (4)

  • Figure 1: Training loss and approximated errors for example 1.
  • Figure 2: Training loss and approximated errors for example 2 with $n=2^9=512$.
  • Figure 3: Model performance with $n=4096$ after training for 5000 epochs for example 1.
  • Figure 4: Model performance with $n=4096$ after training for 5000 epochs for example \ref{['eq:example2']}.

Theorems & Definitions (14)

  • Remark 1
  • Proposition 1
  • proof
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Remark 2
  • Remark 3
  • Proposition 2
  • ...and 4 more