Jul 2026· International journal of multidisciplinary research and analysis· Vol 09· 0 citations
TL;DR
This review examines the evolution of physical modelling from classical first-principles approaches to contemporary data-driven and physics-informed learning frameworks, with a central focus on Physics-Informed Neural Networks and related hybrid methods that integrate governing laws into learning algorithms to improve prediction, generalization, and physical plausibility.
Abstract
This review examines the evolution of physical modelling from classical first-principles approaches to contemporary data-driven and physics-informed learning frameworks. It begins with the historical foundations of scientific modelling, including classical mechanics, field equations, computational simulation, and statistical mechanics, and shows how these traditions established differential equations and numerical methods as core tools in science and engineering. The review then discusses the rise of data-driven modelling, covering statistical learning, machine learning, deep learning, and Gaussian processes, with attention to their strengths in handling complex nonlinear systems and their limitations in interpretability, data requirements, and physical consistency. A central focus is placed on Physics-Informed Neural Networks (PINNs) and related hybrid methods that integrate governing laws into learning algorithms to improve prediction, generalization, and physical plausibility. Applications across fluid dynamics, structural engineering, climate science, biomedicine, materials discovery, and energy systems are highlighted. Finally, the review identifies key challenges related to scalability, uncertainty quantification, robustness, and benchmarking, and outlines future directions such as active learning, operator learning, and scientific foundation models.
The results demonstrate that PINN achieves more accurate and stable full-field vibration reconstructions than conventional PINNs, particularly under conditions involving high-frequency modes, and highlights the potential of hybrid data-physics neural frameworks as an efficient and reliable approach for solving complex PDE-governed dynamical systems.
Hailong Liu, S. Hedayatrasa, Yunpeng Zhu et al.· e-Journal of Nondestructive...· 0 citations
This review examines neural operator architectures for learning solution operators of parametric partial differential equations (PDEs), with an emphasis on conceptual clarity and practical implementation. The work analyzes key models, including DeepONet, PCANet, and the Fourier Neural Operator, highlighting their underlying representations, computational structures, and comparative performance. These architectures are demonstrated on three canonical PDE problems: the Poisson equation, a linear elasticity problem, and a hyperelasticity problem. To make the presentation self-contained, key foundational topics are introduced, including finite-dimensional representations of function spaces, singular-value decomposition, and sampling from infinite-dimensional function spaces. Beyond forward modeling, the review discusses the use of neural operators as surrogate models within a Bayesian inverse-problem framework, including prior specification, forward-map approximation, and posterior computation. The performance of the three neural-operator architectures is evaluated on in-distribution samples, out-of-distribution samples, and Bayesian inference tasks. The review also discusses challenges related to prediction accuracy and generalization, outlining emerging strategies such as residual-based error correction and multi-level training. The review concludes by positioning neural operators within broader scientific-computing workflows and by identifying directions for reliable, scalable operator learning.
Physics‐Informed Neural Networks (PINNs) are deep neural networks that incorporate partial differential equations in their loss function to understand the physics of the system. Thereby enhancing the learning process of the model to achieve better predictive results than the existing models. PINNs embed governing physical principles directly into their loss functions, enabling them to solve complex problems even with limited or noisy data. This distinctive capability allows PINNs to effectively handle complex computational and engineering problems. Such problems include fluid flow prediction, heat transfer simulation, and wave propagation analysis. Hence, a comprehensive analysis of PINNs and their variations for fluid dynamics in the human biological system will be presented in this work. The findings of this numerical evaluation indicate that PINN‐based models are always in agreement with reference and in vivo data, which proves to be more accurate in predicting the major hemodynamic parameters, including velocity, pressure, and wall shear stress (WSS). However, challenges including high computational cost, convergence instability, and limitations in multi‐scale generalization remain significant. Further, the review addresses the foundational architecture of PINNs, improvements introduced through various extensions, and their relative advantages in different biofluid applications. Special emphasis will be placed on their applications in hemodynamics modeling blood flow dynamics and vascular biomechanics using PINNs. This study also discusses the major challenges of PINNs, such as high computational cost, convergence issues, multi‐scale modeling, and limited generalization. It also reviews proposed solutions and suggests future directions, including integration with other machine learning models for improved scalability and broader interdisciplinary applications.
N. R, Amutha S, D. R et al.· WIREs Data Mining and Knowle...· 0 citations
The results demonstrate that the mPINN architecture functions as a reliable, physics-constrained ML framework capable of delivering high-fidelity trajectory predictions for complex multi-body molecular systems.
T. Muther, V. Pham, A. K. Dahaghi· Machine Learning: Science an...· 0 citations
Identifying stochastic dynamical systems from observational data remains a major challenge in applied mathematics and engineering, particularly when complex systems are influenced by random perturbations and incomplete empirical information. This comprehensive review aims to examine state-of-the-art data-driven methods for discovering governing equations, estimating parameters, and predicting the behavior of stochastic dynamical systems. The review systematically analyzes key methodological approaches, including Sparse Identification of Nonlinear Dynamics (SINDy), Dynamic Mode Decomposition (DMD) and its extensions, Koopman operator theory, neural ordinary differential equations, and Bayesian inference. Each approach is evaluated in terms of its theoretical foundations, computational requirements, robustness to noise, and applicability to different classes of stochastic systems. Drawing on numerical experiments and real-world case studies, the findings show that no single method consistently outperforms others across all scenarios. Instead, hybrid approaches that integrate physics-informed constraints with machine learning demonstrate the strongest potential for advancing data-driven system identification. The review concludes that future research should address real-time identification, uncertainty quantification, and the integration of multi-fidelity data sources to improve the reliability and scalability of stochastic system modeling. This work contributes a comprehensive framework for guiding researchers and practitioners in selecting and implementing appropriate identification methods for stochastic dynamical systems.
Rishav Jha, Kameshwar Sahani, S. K. Sahani et al.· African Multidisciplinary Jo...· 0 citations
Physics‐informed neural networks (PINNs) have gained increasing attention in chemical process modeling since they can embed first‐principles knowledge into neural network training. However, in practice, the embedded physics is often inaccurate, and experimental data are costly to obtain, raising fundamental questions about the required physics‐model accuracy and data volume required to achieve a target prediction accuracy. This work develops a theoretical framework for analyzing the generalization error of PINNs under model misspecification. We establish both an architecture‐independent error bound and an explicit bound for a specific PINN architecture. The bound is further related to the solution error with respect to the true system, yielding quantitative conditions on admissible model discrepancy, data requirements, and loss‐weight selection. Based on these conditions, two adaptive algorithms are proposed to guide physics‐model refinement and data collection. The theoretical findings are demonstrated using a chemical process network.
Guoquan Wu, Yuyang Jiang, Yao Shi et al.· AIChE Journal· 0 citations