Vysvětlitelné přístupy umělé inteligence pro řešení stochastických diferenciálních rovnic

Abstract

This thesis presents a symbolic regression approach for solving stochastic differential equations (SDEs) using Grammatical Evolution (GE). The goal is to generate interpretable expressions that approximate the solution trajectories of these equations. Unlike traditional methods such as the Euler-Maruyama scheme, which only provide numerical approximations, the proposed approach aims to produce closed-form symbolic representations of complex stochastic models. Building on the motivation for explainable methods, we first review the GPAD (Genetic Programming and Automatic Differentiation) framework and its use of symbolic modeling for SDEs and their solutions. However, due to practical challenges in implementing this approach, we developed an alternative approach by expanding the Python-based PonyGE2 framework. The proposed method learns symbolic expressions from simulated datasets composed of time and Wiener process values as inputs and the corresponding process values as outputs. We apply this framework to several well-known SDEs, including the Geometric Brownian Motion, Ornstein–Uhlenbeck process, and Cox–Ingersoll–Ross model. Both known and unknown realizations of the Wiener process are considered to assess the accuracy and generalization of the GE approach. To evaluate the performance of the symbolic models, we generate multiple trajectories for each process and compute the mean and variance over time. This statistical comparison helps us to determine how accurately the symbolic models represent the underlying dynamics of the original system. The results demonstrate that GE can recover concise and meaningful symbolic representations that reflect the underlying dynamics of the stochastic systems. This validation confirms that the symbolic models not only fit individual paths but also preserve the statistical properties of the processes. By integrating explainability into the modeling of SDEs, this work offers a new direction for transparent and flexible modeling in stochastic analysis.This thesis presents a symbolic regression approach for solving stochastic differential equations (SDEs) using Grammatical Evolution (GE). The goal is to generate interpretable expressions that approximate the solution trajectories of these equations. Unlike traditional methods such as the Euler-Maruyama scheme, which only provide numerical approximations, the proposed approach aims to produce closed-form symbolic representations of complex stochastic models. Building on the motivation for explainable methods, we first review the GPAD (Genetic Programming and Automatic Differentiation) framework and its use of symbolic modeling for SDEs and their solutions. However, due to practical challenges in implementing this approach, we developed an alternative approach by expanding the Python-based PonyGE2 framework. The proposed method learns symbolic expressions from simulated datasets composed of time and Wiener process values as inputs and the corresponding process values as outputs. We apply this framework to several well-known SDEs, including the Geometric Brownian Motion, Ornstein–Uhlenbeck process, and Cox–Ingersoll–Ross model. Both known and unknown realizations of the Wiener process are considered to assess the accuracy and generalization of the GE approach. To evaluate the performance of the symbolic models, we generate multiple trajectories for each process and compute the mean and variance over time. This statistical comparison helps us to determine how accurately the symbolic models represent the underlying dynamics of the original system. The results demonstrate that GE can recover concise and meaningful symbolic representations that reflect the underlying dynamics of the stochastic systems. This validation confirms that the symbolic models not only fit individual paths but also preserve the statistical properties of the processes. By integrating explainability into the modeling of SDEs, this work offers a new direction for transparent and flexible modeling in stochastic analysis.