Related papers: LawMind: A Law-Driven Paradigm for Discovering Ana…

Modeling nonlinear spatiotemporal dynamical systems has primarily relied on partial differential equations (PDEs). However, the explicit formulation of PDEs for many underexplored processes, such as climate systems, biochemical reaction and…

Solving partial differential equations (PDEs) is a required step in the simulation of natural and engineering systems. The associated computational costs significantly increase when exploring various scenarios, such as changes in initial or…

Partial differential equations (PDEs) play a fundamental role in modeling and simulating problems across a wide range of disciplines. Recent advances in deep learning have shown the great potential of physics-informed neural networks…

Solving Partial Differential Equations (PDEs) is fundamental to numerous scientific and engineering disciplines. A common challenge arises from solving the PDE families, which are characterized by sharing an identical mathematical structure…

Partial differential equations (PDEs) are used to describe a variety of physical phenomena. Often these equations do not have analytical solutions and numerical approximations are used instead. One of the common methods to solve PDEs is the…

Partial Differential Equations (PDEs) are notoriously difficult to solve. In general, closed-form solutions are not available and numerical approximation schemes are computationally expensive. In this paper, we propose to approach the…

Algorithms are presented for the tanh- and sech-methods, which lead to closed-form solutions of nonlinear ordinary and partial differential equations (ODEs and PDEs). New algorithms are given to find exact polynomial solutions of ODEs and…

In recent years, the researches about solving partial differential equations (PDEs) based on artificial neural network have attracted considerable attention. In these researches, the neural network models are usually designed depend on…

Natural laws are often described through differential equations yet finding a differential equation that describes the governing law underlying observed data is a challenging and still mostly manual task. In this paper we make a step…

We introduce Weak-PDE-LEARN, a Partial Differential Equation (PDE) discovery algorithm that can identify non-linear PDEs from noisy, limited measurements of their solutions. Weak-PDE-LEARN uses an adaptive loss function based on weak forms…

Physics-Informed Neural Networks (PINNs) have emerged as a powerful framework for solving partial differential equations (PDEs) by embedding physical laws into neural network training. However, traditional PINN models are typically designed…

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…

Throughout many fields, practitioners often rely on differential equations to model systems. Yet, for many applications, the theoretical derivation of such equations and/or accurate resolution of their solutions may be intractable. Instead,…

Physics-Informed Neural Networks (PINNs) are a novel computational approach for solving partial differential equations (PDEs) with noisy and sparse initial and boundary data. Although, efficient quantification of epistemic and aleatoric…

Reliable predictions of critical phenomena, such as weather, wildfires and epidemics often rely on models described by Partial Differential Equations (PDEs). However, simulations that capture the full range of spatio-temporal scales…

Partial differential equations (PDEs) play a crucial role in studying a vast number of problems in science and engineering. Numerically solving nonlinear and/or high-dimensional PDEs is often a challenging task. Inspired by the traditional…

Deep operator networks (DeepONets) are receiving increased attention thanks to their demonstrated capability to approximate nonlinear operators between infinite-dimensional Banach spaces. However, despite their remarkable early promise,…

Navigating the complexities of physics reasoning has long been a difficult task for Large Language Models (LLMs), requiring a synthesis of profound conceptual understanding and adept problem-solving techniques. In this study, we investigate…

Providing fast and accurate solutions to partial differential equations is a problem of continuous interest to the fields of applied mathematics and physics. With the recent advances in machine learning, the adoption learning techniques in…

Recently, a lot of papers proposed to use neural networks to approximately solve partial differential equations (PDEs). Yet, there has been a lack of flexible framework for convenient experimentation. In an attempt to fill the gap, we…