🤖 AI Summary
Traditional numerical methods struggle to efficiently solve the transient Fokker–Planck equation with arbitrary initial distributions and system parameters in a parallelizable manner, hindering comprehensive parameter-space exploration and transient analysis. This work proposes the Pseudo-Analytical Probabilistic Solution (PAPS) framework, which jointly models solutions across arbitrary multimodal initial conditions, system parameters, and time through a single training procedure. The key innovation lies in unifying initial, transient, and steady-state distributions into a Gaussian mixture representation, employing a constraint-preserving autoencoder to establish a bijective mapping between high-dimensional constrained parameters and a low-dimensional unconstrained latent space, and constructing a physics-informed evolution network to capture global dynamics across varying parameters and initial conditions. Experiments demonstrate that PAPS achieves high accuracy on benchmark systems, with inference speeds four orders of magnitude faster than GPU-accelerated Monte Carlo simulations, enabling real-time parameter sweeps and stochastic bifurcation analysis.
📝 Abstract
Efficiently solving the Fokker-Planck equation (FPE) is central to analyzing complex parameterized stochastic systems. However, current numerical methods lack parallel computation capabilities across varying conditions, severely limiting comprehensive parameter exploration and transient analysis. This paper introduces a deep learning-based pseudo-analytical probability solution (PAPS) that, via a single training process, simultaneously resolves transient FPE solutions for arbitrary multi-modal initial distributions, system parameters, and time points. The core idea is to unify initial, transient, and stationary distributions via Gaussian mixture distributions (GMDs) and develop a constraint-preserving autoencoder that bijectively maps constrained GMD parameters to unconstrained, low-dimensional latent representations. In this representation space, the panoramic transient dynamics across varying initial conditions and system parameters can be modeled by a single evolution network. Extensive experiments on paradigmatic systems demonstrate that the proposed PAPS maintains high accuracy while achieving inference speeds four orders of magnitude faster than GPU-accelerated Monte Carlo simulations. This efficiency leap enables previously intractable real-time parameter sweeps and systematic investigations of stochastic bifurcations. By decoupling representation learning from physics-informed transient dynamics, our work establishes a scalable paradigm for probabilistic modeling of multi-dimensional, parameterized stochastic systems.