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Physics-Informed Neural Networks (PINNs) have emerged as a promising approach for solving Partial Differential Equations (PDEs) by incorporating physical constraints into deep learning models. However, standard PINNs often require a large…

We consider solving the forward and inverse PDEs which have sharp solutions using physics-informed neural networks (PINNs) in this work. In particular, to better capture the sharpness of the solution, we propose adaptive sampling methods…

Recently, researchers have utilized neural networks to accurately solve partial differential equations (PDEs), enabling the mesh-free method for scientific computation. Unfortunately, the network performance drops when encountering a high…

Distributed multi-party learning provides an effective approach for training a joint model with scattered data under legal and practical constraints. However, due to the quagmire of a skewed distribution of data labels across participants…

Physics-informed neural networks (PINNs) and neural operators, two leading scientific machine learning (SciML) paradigms, have emerged as powerful tools for solving partial differential equations (PDEs). Although increasing the training…

Operator learning for time-dependent partial differential equations (PDEs) has seen rapid progress in recent years, enabling efficient approximation of complex spatiotemporal dynamics. However, most existing methods rely on fixed time step…

The approximation of solutions of partial differential equations (PDEs) with numerical algorithms is a central topic in applied mathematics. For many decades, various types of methods for this purpose have been developed and extensively…

Despite considerable scientific advances in numerical simulation, efficiently solving PDEs remains a complex and often expensive problem. Physics-informed Neural Networks (PINN) have emerged as an efficient way to learn surrogate solvers by…

We introduce the Energy Dissipation Rate guided Adaptive Sampling (EDRAS) strategy, a novel method that substantially enhances the performance of Physics-Informed Neural Networks (PINNs) in solving thermodynamically consistent partial…

Physics-informed neural networks (PINNs) have shown promising potential for solving partial differential equations (PDEs) using deep learning. However, PINNs face training difficulties for evolutionary PDEs, particularly for dynamical…

Learning the solution of partial differential equations (PDEs) with a neural network is an attractive alternative to traditional solvers due to its elegance, greater flexibility and the ease of incorporating observed data. However, training…

Solving partial differential equations (PDEs) has been indispensable in scientific and engineering applications. Recently, deep learning methods have been widely used to solve high-dimensional problems, one of which is the physics-informed…

We propose machine learning methods for solving fully nonlinear partial differential equations (PDEs) with convex Hamiltonian. Our algorithms are conducted in two steps. First the PDE is rewritten in its dual stochastic control…

In recent years, with the advancements in machine learning and neural networks, algorithms using physics-informed neural networks (PINNs) to solve PDEs have gained widespread applications. While these algorithms are well-suited for a wide…

This paper introduces an adaptive time splitting technique for the solution of stiff evolutionary PDEs that guarantees an effective error control of the simulation, independent of the fastest physical time scale for highly unsteady…

Autonomous driving vehicles aim to free the hands of vehicle operators, helping them to drive easier and faster, meanwhile, improving the safety of driving on the highway or in complex scenarios. Automated driving systems (ADS) are…

Stochastic gradient decent~(SGD) and its variants, including some accelerated variants, have become popular for training in machine learning. However, in all existing SGD and its variants, the sample size in each iteration~(epoch) of…

In this work, we present a hybrid numerical method for solving evolution partial differential equations (PDEs) by merging the time finite element method with deep neural networks. In contrast to the conventional deep learning-based…

As an alternative to PINNs, a Deep Ritz framework is proposed to solve fully nonlinear PDEs. A least-squares algorithm is advocated to decouple the nonlinearities from the variational features of several fully nonlinear PDEs. A splitting…

Trajectory estimation of maneuvering objects is applied in numerous tasks like navigation, path planning and visual tracking. Many previous works get impressive results in the strictly controlled condition with accurate prior statistics and…