Bifurcation-adaptive physics-informed neural networks for stochastic dynamical systems

The modelling of stochastic dynamical systems that exhibit bifurcations in the vicinity of the edge of chaos poses a critical challenge. These systems encompass two distinct fields of study: materials science, specifically the analysis of thermal stress in silicon crystals, and quantitative finance, with a focus on derivative pricing. It is evident that traditional numerical methods and standard Physics-Informed Neural Networks (PINNs) encounter difficulties when confronted with the coupled effects of Lévy noise, nonlocal diffusion, and abrupt state transitions. This phenomenon gives rise to substantial prediction errors and suboptimal generalisation. In order to address these limitations, this study proposes a Bifurcation-Adaptive PINN (BA-PINN) framework. The model incorporates three innovations: firstly, a bifurcation sensitivity index for precise chaos-edge detection; secondly, a unified stochastic-nonlocal coupling operator; and thirdly, a three-state subnetwork architecture that dynamically adapts to system states via Lyapunov exponent monitoring. The experimental findings demonstrate that BA-PINN reduces thermal stress prediction errors in silicon crystal growth to 0.81–1.10% and decreases dislocation density by 12.1%. In the domain of financial engineering, it attains an option pricing error of 0.43%, reduces hedging costs by 9.3%, and successfully captures 97.1% of extreme market events. Furthermore, the framework reduces ergodicity error to 0.026 and enhances computational efficiency by 3.4 times in comparison with hybrid methods. This study demonstrates the feasibility of the BA-PINN framework on two representative industrial cases: thermal stress prediction in silicon crystal growth and option pricing under extreme volatility. While the results are promising, the framework's applicability to other domains requires further validation. The findings suggest a potential methodological direction for modeling stochastic-nonlocal systems near critical transitions, subject to appropriate domain-specific calibration.