Combining physics-based and data-driven models - advancing the frontiers of research with Scientific Machine Learning

Aug 5, 2026

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Scientific Machine Learning (SciML) combines physics-based and data-driven models to numerically approximate differential problems. Physics-based models start from physical understanding, mathematical formulation, and numerical approximation, while data-driven models learn input-output relations from data without requiring explicit causality assumptions. SciML injects physics and mathematical knowledge into machine learning while exploiting data-driven algorithms to discover complex nonlinear patterns. It surveys foundations and architectures, highlights strategies for PDE-governed problems, and demonstrates successful application to simulation of human cardiac function.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages127

Reading Time5 h 20 min

AccessFree to download

Introduction

Digital models

Mathematical models

Numerical models

Data-driven models

Artificial Intelligence

Machine Learning

A quick glance at Deep Learning models

Scientific Machine Learning

Surrogate modelling of high-fidelity DM

Physics-Informed learning

Operator learning

SciML for the iHeart simulator

The integrated heart model

Multidelity PINNs for ionic parameter estimation

Physics-aware NNs for the inverse problem

Combining physics{based and data{driven models: advancing the frontiers of research with Scienti􀀌c Machine Learning Al􀀌o Quarteroni 1 , Paola Gervasio2 , Francesco Regazzoni3 1 Politecnico di Milano (Professor Emeritus), piazza Leonardo da Vinci, 32, Milan, 20133, Italy and Ecole Polytechnique F􀀓ed􀀓erale de Lausanne (Professor Emeritus), Station 8, Lausanne, 1015, Switzerland [alfio.quarteroni@polimi.it](alfio.quarteroni@polimi.it) 2 DICATAM, University of Brescia, via Branze 43, Brescia, 25123, Italy [paola.gervasio@unibs.it](paola.gervasio@unibs.it) 3 MOX, Department of Mathematics, Politecnico di Milano, piazza Leonardo da Vinci, 32, Milan, 20133, Italy [francesco.regazzoni@polimi.it](francesco.regazzoni@polimi.it) Abstract Scienti􀀌c Machine Learning (SciML) is a recently emerged research 􀀌eld which combines physics{ based and data{driven models for the numerical approximation of di􀀋erential problems. Physics{ based models rely on the physical understanding of the problem at hand, subsequent mathematical formulation, and numerical approximation. Data{driven models instead aim to extract relations between input and output data without arguing any causality principle underlining the available data distribution. In recent years, data{driven models have been rapidly developed and popularized. Such a di􀀋usion has been triggered by a huge availability of data (the so{called big data), an increasingly cheap computing power, and the development of powerful machine learning algorithms. SciML leverages the physical awareness of physics{based models and, at the same time, the e􀀎ciency of data{ driven algorithms. With SciML, we can inject physics and mathematical knowledge into machine learning algorithms. Yet, we can rely on data{driven algorithms' capability to discover complex and non{linear patterns from data and improve the descriptive capacity of physics{based models. After recalling the mathematical foundations of digital modelling and machine learning algorithms, and presenting the most popular machine learning architectures, we discuss the great potential of a broad variety of SciML strategies in solving complex problems governed by partial di􀀋erential equations. Finally, we illustrate the successful application of SciML to the simulation of the human cardiac function, a 􀀌eld of signi􀀌cant socio{economic importance that poses numerous challenges on both the mathematical and computational fronts. The corresponding mathematical model is a complex system of non{linear ordinary and partial di􀀋erential equations describing the electromechanics, valve dynamics, blood circulation, perfusion in the coronary tree, and torso potential. Despite the robustness and accuracy of physics{based models, certain aspects, such as unveiling constitutive laws for cardiac cells and myocardial material properties, as well as devising e􀀎cient reduced order models to dominate the extraordinary computational complexity, have been successfully tackled by leveraging data{driven models. Keywords. Scienti􀀌c Computing, Approximation of PDEs, Machine Learning, Arti􀀌cial Neural Networks, Scienti􀀌c Machine Learning Published in Mathematical Models and Methods in Applied Sciences (M3AS) [https://doi.org/10.1142/](https://doi.org/10.1142/)[ ](https://doi.org/10.1142/)S0218202525500125 Contents 1 Introduction 3 2 Digital models 5 2.1 Mathematical models ....................................... 5 2.2 Numerical models ......................................... 7 3 Data{driven models 11 3.1 Arti􀀌cial Intelligence ....................................... 11 3.2 Machine Learning ......................................... 12 3.2.1 Machine Learning Tasks ................................. 15 3.2.2 Machine Learning Experience .............................. 16 3.2.3 Machine Learning Performance measurement ..................... 16 3.2.4 Machine Learning Models ................................ 17 3.2.5 Setting of supervised learn

FAQs

What distinguishes physics-based from data-driven models in Scientific Machine Learning (SciML)?

Physics-based models rely on physical understanding, mathematical formulation, and numerical approximation. Data-driven models extract relations between inputs and outputs from data without assuming an underlying causality principle for the data distribution.

How does SciML benefit from combining physics-based knowledge with data-driven machine learning?

SciML injects physics and mathematical knowledge into machine learning algorithms, while using data-driven models to discover complex and nonlinear patterns and improve the descriptive capacity of physics-based models.

What areas of machine learning and mathematical foundations does the paper cover before discussing SciML strategies?

The paper recalls mathematical foundations of digital modeling and machine learning algorithms, then presents popular machine learning architectures, including supervised learning setup, error analysis, optimization and backpropagation, plus an overview of deep learning models.

Aug 5, 2026

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