A deep learning approach to solve forward differential problems on graphs
We propose a novel deep learning (DL) approach to solve one-dimensional non-linear elliptic, parabolic, and hyperbolic problems on graphs. A system of physics-informed neural network (PINN) models is used to solve the differential equations, by assigning each PINN model to a specific edge of the graph. Kirkhoff-Neumann (KN) nodal conditions are imposed in a weak form by adding a penalization term to the training loss function. Through the penalization term that imposes the KN conditions, PINN models associated with edges that share a node coordinate with each other to ensure continuity of the solution and of its directional derivatives computed along the respective edges. Using individual PINN models for each edge of the graph allows our approach to fulfill necessary requirements for parallelization by enabling different PINN models to be trained on distributed compute resources. Numerical results show that the system of PINN models accurately approximate the solutions of the differential problems across the entire graph for a broad set of graph topologies.
Code (0)
등록된 구현이 없습니다.
Similar Papers 제목 키워드 기반
A comparison of PINN approaches for drift-diffusion equations on metric graphs
In this paper we focus on comparing machine learning approaches for quantum graphs, which are metric graphs, i.e., graphs with dedicated edge lengths, and an associated differential operator. In our case the differential…
Applications and Manipulations of Physics-Informed Neural Networks in Solving Differential Equations
Mathematical models in neural networks are powerful tools for solving complex differential equations and optimizing their parameters; that is, solving the forward and inverse problems, respectively. A forward problem pre…
Introduction to Solving Quant Finance Problems with Time-Stepped FBSDE and Deep Learning
In this introductory paper, we discuss how quantitative finance problems under some common risk factor dynamics for some common instruments and approaches can be formulated as time-continuous or time-discrete forward-bac…
Deep Learning-Based Least Square Forward-Backward Stochastic Differential Equation Solver for High-Dimensional Derivative Pricing
We propose a new forward-backward stochastic differential equation solver for high-dimensional derivatives pricing problems by combining deep learning solver with least square regression technique widely used in the leas…
regressionDifferential equation and probability inspired graph neural networks for latent variable learning
Probabilistic theory and differential equation are powerful tools for the interpretability and guidance of the design of machine learning models, especially for illuminating the mathematical motivation of learning latent…
Stock Predictiontext-classificationText ClassificationVariational Inference