Abstract
Stochastic differential equations (SDEs) have been widely employed to model the evolution of dynamical systems under uncertainty. They arise in many disciplines such as physics, chemistry, biology and finance. For many realistic models, the system will reach a dynamical equilibrium in the long run, namely, the probability distribution of the system will reach an invariant measure. Sampling invariant distributions from an Itô diffusion process presents a significant challenge in stochastic simulation. Traditional numerical solvers for stochastic differential equations require both a fine step size and a lengthy simulation period, resulting in both biased and correlated samples. Current deep learning-based method solves the stationary Fokker–Planck equation to determine the invariant probability density function in form of deep neural networks, but they generally do not directly address the problem of sampling from the computed density function. Especially when the drift term is nonlinear, the stationary Fokker-Planck equation would be more complicated and has more than one solution.In the first part of this thesis, we introduce a framework that employs a weak generative sampler (WGS) to directly generate independent and identically distributed (iid) samples induced by a transformation map derived from the stationary Fokker–Planck equation. Our proposed loss function is based on the weak form of the Fokker–Planck equation, integrating normalizing flows to characterize the invariant distribution and facilitate sample generation from the base distribution. Our randomized test function circumvents the need for min-max optimization in the traditional weak formulation. Distinct from conventional generative models, our method neither necessitates the computationally intensive calculation of the Jacobian determinant nor the invertibility of the transformation map. A crucial component of our framework is the adaptively chosen family of test functions in the form of Gaussian kernel functions with centres selected from the generated data samples. Experimental results on several benchmark examples demonstrate the effectiveness of our method, which offers both low computational costs and excellent capability in exploring multiple metastable states.
In the second part of this thesis, we extend the weak generative sampler to sample the stationary distribution of McKean-Vlasov processes governed by nonlinear Fokker-Planck equations. To address the nonlinear interaction term in the McKean-Vlasov processes, we propose two methods. The first method is based on the idea of Implicit iteration, where the interaction term in the loss function is handled by directly incorporating the training generative map. The second method employs the concept of Picard iteration, using an approximation of the generative map to linearize the nonlinear interaction term. A variety of numerical examples are presented to demonstrate the effectiveness of the proposed methods.
In the end, we summarize our contribution and discuss our future work.
| Date of Award | 18 Aug 2025 |
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| Original language | English |
| Awarding Institution |
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| Supervisor | Xiang ZHOU (Supervisor) |