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Physics-informed machine learning frameworks such as Physics-Informed Neural Networks (PINNs) and Physics-Informed Extreme Learning Machines (PI-ELMs) have shown great promise for solving partial differential equations (PDEs) but struggle…

We propose a neural network-based meta-learning method to efficiently solve partial differential equation (PDE) problems. The proposed method is designed to meta-learn how to solve a wide variety of PDE problems, and uses the knowledge for…

Physics-informed deep learning often faces optimization challenges due to the complexity of solving partial differential equations (PDEs), which involve exploring large solution spaces, require numerous iterations, and can lead to unstable…

Physics-informed machine learning holds great promise for solving differential equations, yet existing methods struggle with highly oscillatory, multiscale, or singularly perturbed PDEs due to spectral bias, costly backpropagation, and…

To solve high-dimensional parameter-dependent partial differential equations (pPDEs), a neural network architecture is presented. It is constructed to map parameters of the model data to corresponding finite element solutions. To improve…

Partial differential equations (PDEs) are often computationally challenging to solve, and in many settings many related PDEs must be be solved either at every timestep or for a variety of candidate boundary conditions, parameters, or…

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.…

We present a lightweighted neural PDE representation to discover the hidden structure and predict the solution of different nonlinear PDEs. Our key idea is to leverage the prior of ``translational similarity'' of numerical PDE differential…

Many important problems in science and engineering require solving the so-called parametric partial differential equations (PDEs), i.e., PDEs with different physical parameters, boundary conditions, shapes of computation domains, etc.…

The numerical approximation of partial differential equations (PDEs) using neural networks has seen significant advancements through Physics-Informed Neural Networks (PINNs). Despite their straightforward optimization framework and…

In various engineering and applied science applications, repetitive numerical simulations of partial differential equations (PDEs) for varying input parameters are often required (e.g., aircraft shape optimization over many design…

This paper aims to devise an adaptive neural network basis method for numerically solving a second-order semilinear partial differential equation (PDE) with low-regular solutions in two/three dimensions. The method is obtained by combining…

Physics-Informed Neural Networks (PINNs) are machine learning tools that approximate the solution of general partial differential equations (PDEs) by adding them in some form as terms of the loss/cost function of a Neural Network. Most…

Many important problems in science and engineering require solving the so-called parametric partial differential equations (PDEs), i.e., PDEs with different physical parameters, boundary conditions, shapes of computational domains, etc.…

Physics-Informed Neural Network (PINN) has proven itself a powerful tool to obtain the numerical solutions of nonlinear partial differential equations (PDEs) leveraging the expressivity of deep neural networks and the computing power of…

Solving parametric partial differential equations (PDEs) presents significant challenges for data-driven methods due to the sensitivity of spatio-temporal dynamics to variations in PDE parameters. Machine learning approaches often struggle…

We propose a finite-element local basis-based operator learning framework for solving partial differential equations (PDEs). Operator learning aims to approximate mappings from input functions to output functions, where the latter are…

Physics-Informed Neural Networks (PINNs) often suffer from slow convergence, training instability, and reduced accuracy on challenging partial differential equations due to the anisotropic and rapidly varying geometry of their loss…

Foundation models for partial differential equations (PDEs) have emerged as powerful surrogates pre-trained on diverse physical systems, but adapting them to new downstream tasks remains challenging due to limited task-specific data and…

Machine learning for scientific applications faces the challenge of limited data. We propose a framework that leverages a priori known physics to reduce overfitting when training on relatively small datasets. A deep neural network is…