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We consider machine-learning of time-dependent quantities of interest derived from solution trajectories of parabolic partial differential equations. For large-scale or long-time integration scenarios, where using a full order model (FOM)…

Numerical Analysis · Mathematics 2022-05-02 Bernard Haasdonk , Mario Ohlberger , Felix Schindler

Operator learning aims to discover properties of an underlying dynamical system or partial differential equation (PDE) from data. Here, we present a step-by-step guide to operator learning. We explain the types of problems and PDEs amenable…

Numerical Analysis · Mathematics 2025-04-30 Nicolas Boullé , Alex Townsend

Understanding thermal stress evolution in metal additive manufacturing (AM) is crucial for producing high-quality components. Recent advancements in machine learning (ML) have shown great potential for modeling complex multiphysics problems…

Machine Learning · Computer Science 2024-12-30 R. Sharma , Y. B. Guo

This study explores the integration of machine learning (ML) techniques with large eddy simulation (LES) for predicting species mass fraction and flame characteristics in partially premixed turbulent jet flames. The LES simulations,…

Fluid Dynamics · Physics 2025-05-05 Amirali Shateri , Zhiyin Yang , Jianfei Xie

Process optimization for metal additive manufacturing (AM) is crucial to ensure repeatability, control microstructure, and minimize defects. Despite efforts to address this via the traditional design of experiments and statistical process…

Machine Learning · Computer Science 2022-11-18 Susheel Dharmadhikari , Nandana Menon , Amrita Basak

Multiple model reduction techniques have been proposed to tackle linear and non linear problems. Intrusive model order reduction techniques exhibit high accuracy levels, however, they are rarely used as a standalone industrial tool, because…

Computational Engineering, Finance, and Science · Computer Science 2025-04-10 Mikhael Tannous , Chady Ghnatios , Eivind Fonn , Trond Kvamsdal , Francisco Chinesta

Understanding the process of primary and secondary atomization in liquid jets is crucial in describing spray distribution and droplet geometry for industrial applications and is essential in the development of physics-based low-fidelity…

Fluid Dynamics · Physics 2022-05-11 Brendan V Christensen , Mark Owkes

Operator regression provides a powerful means of constructing discretization-invariant emulators for partial-differential equations (PDEs) describing physical systems. Neural operators specifically employ deep neural networks to approximate…

Machine Learning · Computer Science 2023-04-18 Katiana Kontolati , Somdatta Goswami , George Em Karniadakis , Michael D. Shields

Machine Learning (ML) is increasingly used to construct surrogate models for physical simulations. We take advantage of the ability to generate data using numerical simulations programs to train ML models better and achieve accuracy gain…

Computational Physics · Physics 2021-01-29 Paul Novello , Gaël Poëtte , David Lugato , Pietro Congedo

We propose Derivative Learning (DERL), a supervised approach that models physical systems by learning their partial derivatives. We also leverage DERL to build physical models incrementally, by designing a distillation protocol that…

Machine Learning · Computer Science 2026-01-26 Alessandro Trenta , Andrea Cossu , Davide Bacciu

This paper derives predictive reduced-order models for rocket engine combustion dynamics via Operator Inference, a scientific machine learning approach that blends data-driven learning with physics-based modeling. The non-intrusive nature…

Computational Engineering, Finance, and Science · Computer Science 2021-10-19 Shane A. McQuarrie , Cheng Huang , Karen E. Willcox

We treat the accurate simulation of the calcination reaction in particles, where the particles are large and, thus, the inner-particle processes must be resolved. Because these processes need to be described with coupled partial…

Fluid Dynamics · Physics 2023-11-28 Lucas Reineking , Torben Bergold , Enric Illana , Viktor Scherer , Martin Mönnigmann

Partial differential equations (PDEs) underpin the modeling of many natural and engineered systems. It can be convenient to express such models as neural PDEs rather than using traditional numerical PDE solvers by replacing part or all of…

Machine Learning · Computer Science 2025-09-26 Sanket Jantre , Deepak Akhare , Zhiyuan Wang , Xiaoning Qian , Nathan M. Urban

We propose a machine learning-based method to build a system of differential equations that approximates the dynamics of 3D electromechanical models for the human heart, accounting for the dependence on a set of parameters. Specifically,…

Numerical Analysis · Mathematics 2022-04-06 Francesco Regazzoni , Matteo Salvador , Luca Dedè , Alfio Quarteroni

In metals additive manufacturing (AM), materials and components are concurrently made in a single process as layers of metal are fabricated on top of each other in the near-final topology required for the end-use product. Consequently, tens…

Applied Physics · Physics 2020-05-12 N. S. Johnson , P. S. Vulimiri , A. C. To , X. Zhang , C. A. Brice , B. B. Kappes , A. P. Stebner

Ordinary and partial differential equations (ODEs/PDEs) play a paramount role in analyzing and simulating complex dynamic processes across all corners of science and engineering. In recent years machine learning tools are aspiring to…

Machine Learning · Computer Science 2021-06-11 Sifan Wang , Paris Perdikaris

Recent years have witnessed the promise of coupling machine learning methods and physical domain-specific insights for solving scientific problems based on partial differential equations (PDEs). However, being data-intensive, these methods…

Machine Learning · Computer Science 2025-06-03 Wuyang Chen , Jialin Song , Pu Ren , Shashank Subramanian , Dmitriy Morozov , Michael W. Mahoney

The approximation of solutions of partial differential equations (PDEs) with numerical algorithms is a central topic in applied mathematics. For many decades, various types of methods for this purpose have been developed and extensively…

Numerical Analysis · Mathematics 2024-08-26 Lukas Gonon , Arnulf Jentzen , Benno Kuckuck , Siyu Liang , Adrian Riekert , Philippe von Wurstemberger

This work formulates a new approach to reduced modeling of parameterized, time-dependent partial differential equations (PDEs). The method employs Operator Inference, a scientific machine learning framework combining data-driven learning…

Computational Engineering, Finance, and Science · Computer Science 2025-06-16 Shane A McQuarrie , Parisa Khodabakhshi , Karen E Willcox

Reduced-order modeling lies at the interface of numerical analysis and data-driven scientific computing, providing principled ways to compress high-fidelity simulations in science and engineering. We propose a training framework that…

Computational Engineering, Finance, and Science · Computer Science 2026-01-13 Donglin Liu , Francisco García Atienza , Mengwu Guo