We introduce an innovative approach for solving high-dimensional Fokker-Planck-L\'evy (FPL) equations in modeling non-Brownian processes across disciplines such as physics, finance, and ecology. We utilize a fractional score function and Physical-informed neural networks (PINN) to lift the curse of dimensionality (CoD) and alleviate numerical overflow from exponentially decaying solutions with dimensions. The introduction of a fractional score function allows us to transform the FPL equation into a second-order partial differential equation without fractional Laplacian and thus can be readily solved with standard physics-informed neural networks (PINNs). We propose two methods to obtain a fractional score function: fractional score matching (FSM) and score-fPINN for fitting the fractional score function. While FSM is more cost-effective, it relies on known conditional distributions. On the other hand, score-fPINN is independent of specific stochastic differential equations (SDEs) but requires evaluating the PINN model's derivatives, which may be more costly. We conduct our experiments on various SDEs and demonstrate numerical stability and effectiveness of our method in dealing with high-dimensional problems, marking a significant advancement in addressing the CoD in FPL equations.

With the rapid increase of observational, experimental and simulated data for stochastic systems, tremendous efforts have been devoted to identifying governing laws underlying the evolution of these systems. Despite the broad applications of non-Gaussian fluctuations in numerous physical phenomena, the data-driven approaches to extracting stochastic dynamics with L\'{e}vy noise are relatively few. In this work, we propose a Weak Collocation Regression (WCR) to explicitly reveal unknown stochastic dynamical systems, i.e., the Stochastic Differential Equation (SDE) with both $\alpha$-stable L\'{e}vy noise and Gaussian noise, from discrete aggregate data. This method utilizes the evolution equation of the probability distribution function, i.e., the Fokker-Planck (FP) equation. With the weak form of the FP equation, the WCR constructs a linear system of unknown parameters where all integrals are evaluated by Monte Carlo method with the observations. Then, the unknown parameters are obtained by a sparse linear regression. For a SDE with L\'{e}vy noise, the corresponding FP equation is a partial integro-differential equation (PIDE), which contains nonlocal terms, and is difficult to deal with. The weak form can avoid complicated multiple integrals. Our approach can simultaneously distinguish mixed noise types, even in multi-dimensional problems. Numerical experiments demonstrate that our method is accurate and computationally efficient.

This study addresses the challenges in parameter estimation of stochastic differential equations driven by non-Gaussian noises, which are critical in understanding dynamic phenomena such as price fluctuations and the spread of infectious diseases. Previous research highlighted the potential of LSTM networks in estimating parameters of alpha stable Levy driven SDEs but faced limitations including high time complexity and constraints of the LSTM chaining property. To mitigate these issues, we introduce the PEnet, a novel CNN-LSTM-based three-stage model that offers an end to end approach with superior accuracy and adaptability to varying data structures, enhanced inference speed for long sequence observations through initial data feature condensation by CNN, and high generalization capability, allowing its application to various complex SDE scenarios. Experiments on synthetic datasets confirm PEnet significant advantage in estimating SDE parameters associated with noise characteristics, establishing it as a competitive method for SDE parameter estimation in the presence of Levy noise.

Many complex real world phenomena exhibit abrupt, intermittent or jumping behaviors, which are more suitable to be described by stochastic differential equations under non-Gaussian L\'evy noise. Among these complex phenomena, the most likely transition paths between metastable states are important since these rare events may have a high impact in certain scenarios. Based on the large deviation principle, the most likely transition path could be treated as the minimizer of the rate function upon paths that connect two points. One of the challenges to calculate the most likely transition path for stochastic dynamical systems under non-Gaussian L\'evy noise is that the associated rate function can not be explicitly expressed by paths. For this reason, we formulate an optimal control problem to obtain the optimal state as the most likely transition path. We then develop a neural network method to solve this issue. Several experiments are investigated for both Gaussian and non-Gaussian cases.

The mean exit time escaping basin of attraction in the presence of white noise is of practical importance in various scientific fields. In this work, we propose a strategy to control mean exit time of general stochastic dynamical systems to achieve a desired value based on the quasipotential concept and machine learning. Specifically, we develop a neural network architecture to compute the global quasipotential function. Then we design a systematic iterated numerical algorithm to calculate the controller for a given mean exit time. Moreover, we identify the most probable path between metastable attractors with help of the effective Hamilton-Jacobi scheme and the trained neural network. Numerical experiments demonstrate that our control strategy is effective and sufficiently accurate.

From this point of view, dynamical modeling requires a deep understanding of the process to be analyzed. The essence of model abstraction is an approximation to the observed reality, which is usually represented by a system composed of ordinary or partial differential equations, deterministic or stochastic differential equations, and control equations. Although mathematical models are accurate for many processes, it is particularly difficult to develop such models for some of the most challenging systems, including climate dynamics, brain dynamics, biological systems and financial markets. Fortunately, more and more data are observed or measured in recent years with the development of scientific tools and simulation capabilities. Therefore, a large number of data-driven methods has been proposed to discover governing laws of systems from data. For instance, several researchers designed the Sparse Identification of Nonlinear Dynamics approach to extract deterministic ordinary [5] or partial [15, 29, 31] differential equations from available path data.

The Fokker-Planck (FP) equation governing the evolution of the probability density function (PDF) is applicable to many disciplines but it requires specification of the coefficients for each case, which can be functions of space-time and not just constants, hence requiring the development of a data-driven modeling approach. When the data available is directly on the PDF, then there exist methods for inverse problems that can be employed to infer the coefficients and thus determine the FP equation and subsequently obtain its solution. Herein, we address a more realistic scenario, where only sparse data are given on the particles' positions at a few time instants, which are not sufficient to accurately construct directly the PDF even at those times from existing methods, e.g., kernel estimation algorithms. To this end, we develop a general framework based on physics-informed neural networks (PINNs) that introduces a new loss function using the Kullback-Leibler divergence to connect the stochastic samples with the FP equation, to simultaneously learn the equation and infer the multi-dimensional PDF at all times. In particular, we consider two types of inverse problems, type I where the FP equation is known but the initial PDF is unknown, and type II in which, in addition to unknown initial PDF, the drift and diffusion terms are also unknown. In both cases, we investigate problems with either Brownian or Levy noise or a combination of both. We demonstrate the new PINN framework in detail in the one-dimensional case (1D) but we also provide results for up to 5D demonstrating that we can infer both the FP equation and} dynamics simultaneously at all times with high accuracy using only very few discrete observations of the particles.

With the rapid increase of valuable observational, experimental and simulating data for complex systems, great efforts are being devoted to discovering governing laws underlying the evolution of these systems. However, the existing techniques are limited to extract governing laws from data as either deterministic differential equations or stochastic differential equations with Gaussian noise. In the present work, we develop a new data-driven approach to extract stochastic dynamical systems with non-Gaussian symmetric L\'evy noise, as well as Gaussian noise. First, we establish a feasible theoretical framework, by expressing the drift, diffusion coefficient and jump measure (i.e., anomalous diffusion) for the underlying stochastic dynamical system in terms of sample paths data. We then design a numerical algorithm to compute the drift, diffusion coefficient and jump measure, and thus extract a governing stochastic differential equation with Gaussian and non-Gaussian noise. Finally, we demonstrate the efficacy and accuracy of our approach by applying to several prototypical one-, two- and three-dimensional systems. This new approach will become a tool in discovering governing dynamical laws from noisy data sets, from observing or simulating complex phenomena, such as rare events triggered by random fluctuations with heavy as well as light tail statistical features.