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Physics-informed neural networks (PINNs) have attracted significant attention for solving partial differential equations (PDEs) in recent years because they alleviate the curse of dimensionality that appears in traditional methods. However,…

In recent years, deep learning technology has been used to solve partial differential equations (PDEs), among which the physics-informed neural networks (PINNs) emerges to be a promising method for solving both forward and inverse PDE…

This work presents a physics-informed deep learning-based super-resolution framework to enhance the spatio-temporal resolution of the solution of time-dependent partial differential equations (PDE). Prior works on deep learning-based…

This work proposes an extension of neural ordinary differential equations (NODEs) by introducing an additional set of ODE input parameters to NODEs. This extension allows NODEs to learn multiple dynamics specified by the input parameter…

Neural PDE solvers are increasingly used as learned surrogates for families of partial differential equations, where the key machine learning challenge is not only interpolation on a fixed benchmark distribution but generalization under…

Polynomial chaos expansion (PCE) is a classical and widely used surrogate modeling technique in physical simulation and uncertainty quantification. By taking a linear combination of a set of basis polynomials - orthonormal with respect to…

Multiphysics problems such as multicomponent diffusion, phase transformations in multiphase systems and alloy solidification involve numerical solution of a coupled system of nonlinear partial differential equations (PDEs). Numerical…

Learning dynamics governed by differential equations is crucial for predicting and controlling the systems in science and engineering. Neural Ordinary Differential Equation (NODE), a deep learning model integrated with differential…

Deep learning has become a pivotal technology in fields such as computer vision, scientific computing, and dynamical systems, significantly advancing these disciplines. However, neural Networks persistently face challenges related to…

Machine learning based partial differential equations (PDEs) solvers have received great attention in recent years. Most progress in this area has been driven by deep neural networks such as physics-informed neural networks (PINNs) and…

Data-driven discovery of partial differential equations (PDEs) has attracted increasing attention in recent years. Although significant progress has been made, certain unresolved issues remain. For example, for PDEs with high-order…

The predictive accuracy of operator learning frameworks depends on the quality and quantity of available training data (input-output function pairs), often requiring substantial amounts of high-fidelity data, which can be challenging to…

Numerical methods such as finite element have been flourishing in the past decades for modeling solid mechanics problems via solving governing partial differential equations (PDEs). A salient aspect that distinguishes these numerical…

Physics-informed neural network (PINN) is a powerful emerging method for studying forward-inverse problems of partial differential equations (PDEs), even from limited sample data. Variable coefficient PDEs, which model real-world phenomena,…

Learning solution operators of partial differential equations (PDEs) from data has emerged as a promising route to fast surrogate models in multi-query scientific workflows. However, for geometric PDEs whose inputs and outputs transform…

Across numerous applications, forecasting relies on numerical solvers for partial differential equations (PDEs). Although the use of deep-learning techniques has been proposed, actual applications have been restricted by the fact the…

Learning PDE dynamics from limited data with unknown physics is challenging. Existing neural PDE solvers either require large datasets or rely on known physics (e.g., PDE residuals or handcrafted stencils), leading to limited applicability.…

Time-dependent partial differential equations (PDEs) are ubiquitous in science and engineering. Recently, mostly due to the high computational cost of traditional solution techniques, deep neural network based surrogates have gained…

Nonlinear Partial Differential Equations (PDEs) are ubiquitous in mathematical physics and engineering. Although Physics-Informed Neural Networks (PINNs) have emerged as a powerful tool for solving PDE problems, they typically struggle to…

Partial differential equations (PDEs) play a crucial role in financial mathematics, particularly in portfolio optimization, and solving them using classical numerical or neural network methods has always posed significant challenges. Here,…