Neural Operator - Is data all you need to model the world - An insight into the impact of Physics Informed Machine Learning

Aug 5, 2026

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Numerical approximations of partial differential equations underpin solutions to heat and sound propagation, fluid flow, elasticity, electrostatics, electrodynamics, and related physics and engineering problems. Traditional finite element, finite difference, and finite volume methods are accurate but computationally expensive. Data-driven neural approaches offer discretization and resolution invariance, but introduce key failure modes. The work surveys how data-driven techniques complement conventional solvers and highlights a fast ~1000x neural-operator learning strategy for PDE operators, enabling efficient modeling in fundamental and applied physics.

LanguageEnglish

CategoryResearch & Report

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Pages25

Reading Time1 h 3 min

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Introduction

Why PDEs matter in physics and engineering

Conventional numerical methods: FDM, FEM, FVM

Data-driven machine learning alternatives

Neural Operator: Is data all you need to model the world? An insight into the impact of Physics Informed Machine Learning Hrishikesh Viswanatha,∗, Md Ashiqur Rahmana , Abhijeet Vyasa , Andrey Shora , Beatriz Medeirosd , Stephanie Hernandezd , Suhas Eswarappa Prameelab,c,d , Aniket Beraa a Department of Computer Science, Purdue University, West Lafayette, IN, USA b Department of Materials Science and Engineering, MIT, Cambridge, MA, USA c Department of Aeronautics and Astronautics, MIT, Cambridge, MA, USA d Hopkins Extreme Materials Institute, Johns Hopkins University, Baltimore, MD, USA Abstract Numerical approximations of partial differential equations (PDEs) are routinely employed to formulate the solution of physics, engineering and mathematical problems involving functions of several variables, such as the propagation of heat or sound, fluid flow, elasticity, electrostatics, electrodynamics, and more. While this has led to solving many complex phenomena, there are some limitations. Conventional approaches such as Finite Element Methods (FEMs) and Finite Differential Methods (FDMs) require considerable time and are computationally expensive. In contrast, data driven machine learning-based methods such as neural networks provide a faster, fairly accurate alternative, and have certain advantages such as discretization invariance and resolution invariance. This article aims to provide a comprehensive insight into how data-driven approaches can complement conventional techniques to solve engineering and physics problems, while also noting some of the major pitfalls of machine learning-based approaches. Furthermore, we highlight, a novel and fast machine learning-based approach ( ∼ 1000x) to learning the solution operator of a PDE operator learning. We will note how these new computational approaches can bring immense advantages in tackling many problems in fundamental and applied physics. Keywords: Machine learning, Neural networks, Neural operators, Fourier neural operator, Geo-FNO, Graph neural operator, Physics informed neural operator, Finite element method, Finite volume method, Finite difference method, DeepONet, Spectral neural operator, Adaptive Fourier neural operator, Burgers equation, Darcy Flow equation, Navier Stokes equation, Kolmogorov Flow 1. Introduction Partial differential equations (PDEs) are an integral tool in mathematically modeling the physical world. They allow one to describe how a quantity changes with respect to multiple variables and have allowed physicists to model various phenomena in fluid flow, electrodynamics, and quantum mechanics. An example family of generic PDEs can be represented as shown in equation 1 , (Lau)(x) = f(x), x ∈ D, (1) to solving PDEs are numerical methods such as finite difference methods (FDMs) Godunov & Bohachevsky (1959), finite element methods (FEMs) Zienkiewicz et al.(2005), and finite volume methods (FVMs) Eymardet al. (2000) as they are able to approximate solutions to PDEs with high amounts of accuracy. However, they are computationally expensive. Finite difference methods solve PDEs by converting them into linear algebraic equations called finite difference equations. These equations are obtained by discretizing arXiv :2301 . 13331v2 [ cs .AI] 18 Sep 2023 u (x) = 0, x ∈ δD for some a ∈ A, f ∈ L, where A, L are Banach spaces, Dis the domain of the PDE and u : D → R, u ∈ U is the solution function. While PDEs are all around us, it is oftentimes very difficult for one to solve them analytically. The best that one can achieve is an approximation of the true solution of the PDE. The most popular approaches ∗ Corresponding author Email address: [hviswan@purdue.edu](hviswan@purdue.edu) (Hrishikesh Viswanath) the domain of the functions involved in the PDEs and representing the derivatives as differences according to the first principles of calculus Jordan & Jord´an (1965) . The different discretization schemes result in different meth

FAQs

Why are neural-operator approaches proposed for PDE problems?

Conventional numerical solvers are computationally expensive, while data-driven methods can provide faster approximations and can offer discretization and resolution invariance.

What limitations are associated with conventional PDE solvers like FEM, FDM, and FVM?

They require considerable time and can be computationally expensive, despite their ability to approximate PDE solutions with high accuracy.

How does the article relate physics-informed machine learning to learning PDE operators?

It discusses how data-driven approaches complement conventional techniques, highlights pitfalls, and emphasizes a novel fast method (~1000x) for learning the solution operator of a PDE operator learning task.

Aug 5, 2026

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