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In this study, we present and validate the predictive capability of the Physics-Informed Neural Networks (PINNs) methodology for solving a variety of engineering and biological dynamical systems governed by ordinary differential equations…

Physics-informed neural networks (PINNs) provide a promising machine learning framework for solving partial differential equations, but their training often breaks down on challenging problems, sometimes converging to physically incorrect…

Physics-Informed Neural Networks (PINNs) have enabled significant improvements in modelling physical processes described by partial differential equations (PDEs). PINNs are based on simple architectures, and learn the behavior of complex…

Physics-informed neural networks (PINNs) have emerged as a promising approach to solving partial differential equations (PDEs) using neural networks, particularly in data-scarce scenarios, due to their unsupervised training capability.…

Traditional data-driven deep learning models often struggle with high training costs, error accumulation, and poor generalizability in complex physical processes. Physics-informed deep learning (PiDL) addresses these challenges by…

Physics-informed neural networks (PINNs) as a means of discretizing partial differential equations (PDEs) are garnering much attention in the Computational Science and Engineering (CS&E) world. At least two challenges exist for PINNs at…

Solving partial differential equations (PDEs) by numerical methods meet computational cost challenge for getting the accurate solution since fine grids and small time steps are required. Machine learning can accelerate this process, but…

This paper proposes two efficient approximation methods to solve high-dimensional fully nonlinear partial differential equations (NPDEs) and second-order backward stochastic differential equations (2BSDEs), where such high-dimensional fully…

Physics-Informed Neural Networks (PINNs) have emerged as powerful tools for solving partial differential equations (PDEs). However, training PINNs from scratch is often computationally intensive and time-consuming. To address this problem,…

Learning the full family of solutions to parameterized partial differential equations (PDEs) is a central challenge to our ability to model the behavior of heterogeneous systems, with a variety of fundamental and application-oriented…

The backward Euler-Maruyama (BEM) method is employed to approximate the invariant measure of stochastic differential equations, where both the drift and the diffusion coefficient are allowed to grow super-linearly. The existence and…

Symmetries have been leveraged to improve the generalization of neural networks through different mechanisms from data augmentation to equivariant architectures. However, despite their potential, their integration into neural solvers for…

In this paper, we investigate the problem of strong approximation of the solutions of stochastic differential equations (SDEs) when the drift coefficient is given in integral form. We investigate its upper error bounds, in terms of the…

For stochastic differential equations (SDEs) with Markovian switching, whose drift and diffusion coefficients are allowed to contain superlinear terms, the backward Euler-Maruyama (BEM) method is proposed to approximate the invariant…

Neural SDEs combine many of the best qualities of both RNNs and SDEs: memory efficient training, high-capacity function approximation, and strong priors on model space. This makes them a natural choice for modelling many types of temporal…

We propose a Bayesian physics-informed neural network (B-PINN) to solve both forward and inverse nonlinear problems described by partial differential equations (PDEs) and noisy data. In this Bayesian framework, the Bayesian neural network…

Physics-informed neural networks (PINNs) have emerged as a new learning paradigm for solving partial differential equations (PDEs) by enforcing the constraints of physical equations, boundary conditions (BCs), and initial conditions (ICs)…

Neural Stochastic Differential Equations (Neural SDEs) have emerged as powerful mesh-free generative models for continuous stochastic processes, with critical applications in fields such as finance, physics, and biology. Previous…

A physics-informed neural network (PINN) uses physics-augmented loss functions, e.g., incorporating the residual term from governing partial differential equations (PDEs), to ensure its output is consistent with fundamental physics laws.…

The numerical solution of high dimensional partial differential equations (PDEs) is severely constrained by the curse of dimensionality (CoD), rendering classical grid--based methods impractical beyond a few dimensions. In recent years,…