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The discovery of Partial Differential Equations (PDEs) is an essential task for applied science and engineering. However, data-driven discovery of PDEs is generally challenging, primarily stemming from the sensitivity of the discovered…

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…

Robust physics discovery is of great interest for many scientific and engineering fields. Inspired by the principle that a representative model is the one simplest possible, a new model selection criteria considering both model's Parsimony…

Many problems in science and engineering can be represented by a set of partial differential equations (PDEs) through mathematical modeling. Mechanism-based computation following PDEs has long been an essential paradigm for studying topics…

In this work, we present an adjoint-based method for discovering the underlying governing partial differential equations (PDEs) given data. The idea is to consider a parameterized PDE in a general form and formulate a PDE-constrained…

Pattern formation is a widely observed phenomenon in diverse fields including materials physics, developmental biology and ecology, among many others. The physics underlying the patterns is specific to the mechanisms, and is encoded by…

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…

Efficient and stable solution of partial differential equations (PDEs) is central to scientific and engineering applications, yet existing numerical solvers rely heavily on matrix based discretizations, while learning based methods require…

Recent advancements in machine learning have transformed the discovery of physical laws, moving from manual derivation to data-driven methods that simultaneously learn both the structure and parameters of governing equations. This shift…

In this paper, we present an initial attempt to learn evolution PDEs from data. Inspired by the latest development of neural network designs in deep learning, we propose a new feed-forward deep network, called PDE-Net, to fulfill two…

Data-driven discovery of partial differential equations (PDEs) offers a promising paradigm for uncovering governing physical laws from observational data. However, in practical scenarios, measurements are often contaminated by noise and…

We propose a novel composite framework to find unknown fields in the context of inverse problems for partial differential equations (PDEs). We blend the high expressibility of deep neural networks as universal function estimators with the…

Modeling complex spatiotemporal dynamical systems, such as the reaction-diffusion processes, have largely relied on partial differential equations (PDEs). However, due to insufficient prior knowledge on some under-explored dynamical…

Embedding physical knowledge into neural network (NN) training has been a hot topic. However, when facing the complex real-world, most of the existing methods still strongly rely on the quantity and quality of observation data. Furthermore,…

We introduce a practical method to enforce partial differential equation (PDE) constraints for functions defined by neural networks (NNs), with a high degree of accuracy and up to a desired tolerance. We develop a differentiable…

Effectively controlling systems governed by Partial Differential Equations (PDEs) is crucial in several fields of Applied Sciences and Engineering. These systems usually yield significant challenges to conventional control schemes due to…

Data-driven methods have recently been developed to discover underlying partial differential equations (PDEs) of physical problems. However, for these methods, a complete candidate library of potential terms in a PDE are usually required.…

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…

The data-driven recovery of the unknown governing equations of dynamical systems has recently received an increasing interest. However, the identification of governing equations remains challenging when dealing with noisy and partial…

Discovering governing equations that describe complex chaotic systems remains a fundamental challenge in physics and neuroscience. Here, we introduce the PEM-UDE method, which combines the prediction-error method with universal differential…