MATHEMATICAL MODELING OF STOCHASTIC DYNAMICAL SYSTEMS IN FINANCE BASED ON NEURAL OPERATORS AND PHYSICS-INFORMED LEARNING: AN ANALYTICAL REVIEW

Dzhalmukhambetova E., Kartashov M., Syachina E., Toshpulotov A.

A systematic analysis of machine learning approaches for solving SDEs in quantitative finance is conducted. The study compares classical numerical methods, physics-informed neural networks (PINNs), and operator architectures (DeepONet, FNO). It is established that neural operators overcome the "curse of dimensionality," providing multi-order computational acceleration for calibration and pricing tasks in dimensions d > 50. Key limitations are identified: the retraining intensity of PINNs and the high data dependency of operator methods. The study substantiates the effectiveness of hybrid physics-informed neural operators (PINO) for multi-asset portfolio risk management, as they combine high-speed inference with the physical consistency of the model.

Keywords:neural network operators, physics-informed neural networks, stochastic differential equations, derivative pricing, curse of dimensionality, model calibration, quantitative finance