Neural Laplace for learning Stochastic Differential Equations
Adrien Carrel
TL;DR
The paper extends Neural Laplace to stochastic differential equations by defining a Laplace-domain representation $F(s)=\mathcal{L}\{X\}(s)$ for stochastic processes and deriving GBM-specific mean and variance bounds. It demonstrates that under certain conditions (e.g., small $|X_0|$ and large drift $\mu$), $F(s)$ is well-defined and learnable, enabling neural estimation of SDE dynamics. Empirically, it applies the framework to Geometric Brownian Motion and compares against several ODE-focused baselines, showing competitive performance and validating the feasibility of a unified Laplace-domain approach to diverse DEs. The work highlights potential for broader SDE classes, uncertainty-aware extensions, and integration with neural SDE methods for enhanced stochastic dynamics learning.
Abstract
Neural Laplace is a unified framework for learning diverse classes of differential equations (DE). For different classes of DE, this framework outperforms other approaches relying on neural networks that aim to learn classes of ordinary differential equations (ODE). However, many systems can't be modelled using ODEs. Stochastic differential equations (SDE) are the mathematical tool of choice when modelling spatiotemporal DE dynamics under the influence of randomness. In this work, we review the potential applications of Neural Laplace to learn diverse classes of SDE, both from a theoretical and a practical point of view.