Score-fPINN: Fractional Score-Based Physics-Informed Neural Networks for High-Dimensional Fokker-Planck-Levy Equations

• URL: http://arxiv.org/abs/2406.11676v1
• Date: Mon, 17 Jun 2024 15:57:23 GMT
• Title: Score-fPINN: Fractional Score-Based Physics-Informed Neural Networks for High-Dimensional Fokker-Planck-Levy Equations
• Authors: Zheyuan Hu, Zhongqiang Zhang, George Em Karniadakis, Kenji Kawaguchi,
• Abstract summary: We introduce an innovative approach for solving high-dimensional Fokker-Planck-L'evy (FPL) equations in modeling non-Brownian processes. 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.
• Score: 24.86574584293979
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

Related papers

• Solving Poisson Equations using Neural Walk-on-Spheres [80.1675792181381]
  We propose Neural Walk-on-Spheres (NWoS), a novel neural PDE solver for the efficient solution of high-dimensional Poisson equations. We demonstrate the superiority of NWoS in accuracy, speed, and computational costs.
  arXiv  Detail & Related papers  (2024-06-05T17:59:22Z)
• Score-Based Physics-Informed Neural Networks for High-Dimensional Fokker-Planck Equations [27.164040990410065]
  We propose a score-based solver to fit the score function in SDEs. The proposed score-based SDE solver operates in two stages: first, employing SM, SSM, or Score-PINN to acquire the score; and second, solving the LL via an ODE. The numerical results demonstrate the score-based SDE solver's stability, speed, and performance across different settings.
  arXiv  Detail & Related papers  (2024-02-12T07:59:25Z)
• Deep Equilibrium Based Neural Operators for Steady-State PDEs [100.88355782126098]
  We study the benefits of weight-tied neural network architectures for steady-state PDEs. We propose FNO-DEQ, a deep equilibrium variant of the FNO architecture that directly solves for the solution of a steady-state PDE.
  arXiv  Detail & Related papers  (2023-11-30T22:34:57Z)
• Physics-Informed Solution of The Stationary Fokker-Plank Equation for a Class of Nonlinear Dynamical Systems: An Evaluation Study [0.0]
  An exact analytical solution of the Fokker-Planck (FP) equation is only available for a limited subset of dynamical systems. To evaluate its potential, we present a data-free, physics-informed neural network (PINN) framework to solve the FP equation.
  arXiv  Detail & Related papers  (2023-09-25T13:17:34Z)
• D4FT: A Deep Learning Approach to Kohn-Sham Density Functional Theory [79.50644650795012]
  We propose a deep learning approach to solve Kohn-Sham Density Functional Theory (KS-DFT) We prove that such an approach has the same expressivity as the SCF method, yet reduces the computational complexity. In addition, we show that our approach enables us to explore more complex neural-based wave functions.
  arXiv  Detail & Related papers  (2023-03-01T10:38:10Z)
• Efficient physics-informed neural networks using hash encoding [0.0]
  Physics-informed neural networks (PINNs) have attracted a lot of attention in scientific computing. We propose to incorporate multi-resolution hash encoding into PINNs to improve the training efficiency. We test the proposed method on three problems, including Burgers equation, Helmholtz equation, and Navier-Stokes equation.
  arXiv  Detail & Related papers  (2023-02-26T20:00:23Z)
• Neural Basis Functions for Accelerating Solutions to High Mach Euler Equations [63.8376359764052]
  We propose an approach to solving partial differential equations (PDEs) using a set of neural networks. We regress a set of neural networks onto a reduced order Proper Orthogonal Decomposition (POD) basis. These networks are then used in combination with a branch network that ingests the parameters of the prescribed PDE to compute a reduced order approximation to the PDE.
  arXiv  Detail & Related papers  (2022-08-02T18:27:13Z)
• Monte Carlo PINNs: deep learning approach for forward and inverse problems involving high dimensional fractional partial differential equations [8.378422134042722]
  We introduce a sampling based machine learning approach, Monte Carlo physics informed neural networks (MC-PINNs) for solving forward and inverse fractional partial differential equations (FPDEs) As a generalization of physics informed neural networks (PINNs), our method relies on deep neural network surrogates in addition to an approximation strategy for computing the fractional derivatives of the outputs. We validate the performance of MC-PINNs via several examples that include high dimensional integral fractional Laplacian equations, parametric identification of time-space fractional PDEs, and fractional diffusion equation with random inputs.
  arXiv  Detail & Related papers  (2022-03-16T09:52:05Z)
• Learning Physics-Informed Neural Networks without Stacked Back-propagation [82.26566759276105]
  We develop a novel approach that can significantly accelerate the training of Physics-Informed Neural Networks. In particular, we parameterize the PDE solution by the Gaussian smoothed model and show that, derived from Stein's Identity, the second-order derivatives can be efficiently calculated without back-propagation. Experimental results show that our proposed method can achieve competitive error compared to standard PINN training but is two orders of magnitude faster.
  arXiv  Detail & Related papers  (2022-02-18T18:07:54Z)
• Physics-Informed Neural Operator for Learning Partial Differential Equations [55.406540167010014]
  PINO is the first hybrid approach incorporating data and PDE constraints at different resolutions to learn the operator. The resulting PINO model can accurately approximate the ground-truth solution operator for many popular PDE families.
  arXiv  Detail & Related papers  (2021-11-06T03:41:34Z)
• Solving inverse-PDE problems with physics-aware neural networks [0.0]
  We propose a novel framework to find unknown fields in the context of inverse problems for partial differential equations. We blend the high expressibility of deep neural networks as universal function estimators with the accuracy and reliability of existing numerical algorithms.
  arXiv  Detail & Related papers  (2020-01-10T18:46:50Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.