Physics-informed Neural Networks Research Articles

Physics-informed neural networks and higher-order high-resolution methods for resolving discontinuities and shocks: A comprehensive study

Addressing discontinuities in fluid flow problems is inherently difficult, especially when shocks arise due to the nonlinear nature of the flow. While handling discontinuities is a well-established practice in computational fluid dynamics (CFD), it remains a major challenge when applying physics-informed neural networks (PINNs). In this study, we compare the shock-resolving capabilities of traditional CFD methods with those of PINNs, highlighting the advantages of the latter. Our findings show that PINNs exhibit less dissipative behavior compared to conventional techniques. We evaluated the performance of both PINNs and traditional methods on linear and nonlinear test cases, demonstrating that PINNs offer superior shock-resolving properties. Notably, PINNs can accurately resolve inviscid shocks with just three grid points, whereas traditional methods require at least seven points. This suggests that PINNs are more effective at resolving shocks and discontinuities when using the same grid for both PINN and CFD simulations. However, it’s important to note that PINNs, in this context, are computationally more expensive than traditional methods on a given grid.

Recurrent neural networks as a physics-based self-learning solver to satisfy plane stress viscoplasticity undergoing isotropic damage

The present study introduces a new self-learning framework based on a recurrent neural network to compute the nonlinear viscoplastic response of plate structures undergoing isotropic damage. The neural network comprising Legendre Memory Unit (LMU) cells and dense transformations is pretrained with a combination of data-driven and physics-based loss functions. Once pretrained, the model is deployed in the explicit integration scheme to predict the strain increment in the thickness direction satisfying plane stress condition. This results in elimination/reduction in the iterations required to fulfill the plane stress condition and leads to faster convergence since the neural network replaces the root-finding methods. The proposed self-learning method differentiates itself from the well-known Physics-Informed Neural Networks (PINNs) since the neural network model self-learns online through the proposed physics-based loss term without the need for data. We compare the results of the Neural Network-enhanced integration scheme to validate the proposed method with those of the classical explicit integration scheme.

PINN for solving forward and inverse problems involving integrable two-dimensional nonlocal equations

In this paper, the physics informed neural network (PINN) deep learning method is applied to solve two-dimensional nonlocal equations, including the partial reverse space y-nonlocal Mel’nikov equation, the partial reverse space-time nonlocal Mel’nikov equation and the nonlocal two-dimensional nonlinear Schrödinger (NLS) equation. By the PINN method, we successfully derive a data-driven two soliton solution, lump solution and rogue wave solution. Numerical simulation results indicate that the error range between the data-driven solution and the exact solution is relatively small, which verifies the effectiveness of the PINN deep learning method for solving high dimensional nonlocal equations. Moreover, the parameter discovery of the partial reverse space-time nonlocal Mel’nikov equation is analysed in terms of its soliton solution for the first time.

Advancements of Exploiting Convolutional Neural Networks for Solving Differential Equations

Solving Partial Differential Equations (PDEs) is essential across various fields, including physics, mathematics, and engineering. This study explores innovative methods for solving PDEs using Convolutional Neural Networks (CNNs). Traditionally, solving PDEs through numerical methods like finite difference or spectral approaches is computationally intensive, particularly when addressing high-dimensional problems and complex boundary conditions. CNNs, with their ability to handle spatial data efficiently, offer a promising alternative. This research evaluates different neural network architectures, including data-driven, physics-driven, and hybrid approaches, and their role in enhancing the accuracy and scalability of PDE solutions. Data-driven methods rely on extensive datasets for training neural networks, while Physics-Informed Neural Networks (PINNs) integrate the underlying physical laws into the solution process. Hybrid approaches combine both strategies, leveraging data and physical principles. The paper emphasizes CNNs potential to transform traditional PDE-solving methods, making them more efficient and adaptable to complex, real-world applications. Future research directions include optimizing these methods for greater scalability and interdisciplinary applications.

Hybrid CFD PINN FSI Simulation in Coronary Artery Trees

This paper presents a novel hybrid approach that integrates computational fluid dynamics (CFD), physics-informed neural networks (PINN), and fluid–structure interaction (FSI) methods to simulate fluid flow in stenotic coronary artery trees and predict fractional flow reserve (FFR) in areas of stenosis. The primary objective is to utilize a 1D PINN model to accurately predict outlet flow conditions, effectively addressing the challenges of measuring or estimating these conditions within complex arterial networks. Validation against traditional CFD methods demonstrates strong accuracy while embedding physics-based training to ensure compliance with fundamental fluid dynamics principles. The findings indicate that the hybrid CFD PINN FSI method generates realistic outflow boundary conditions crucial for diagnosing stenosis, requiring minimal input data. By seamlessly integrating initial conditions established by the 1D PINN into FSI simulations, this approach enables precise assessments of blood flow dynamics and FFR values in stenotic regions. This innovative application of 1D PINN not only distinguishes this methodology from conventional data-driven models that rely heavily on extensive datasets but also highlights its potential to enhance our understanding of hemodynamics in pathological states. Ultimately, this research paves the way for significant advancements in non-invasive diagnostic techniques in cardiology, improving clinical decision making and patient outcomes.

Optimising Physics-Informed Neural Network Solvers for Turbulence Modelling: A Study on Solver Constraints Against a Data-Driven Approach

Physics-informed neural networks (PINNs) have emerged as a promising approach for simulating nonlinear physical systems, particularly in the field of fluid dynamics and turbulence modelling. Traditional turbulence models often rely on simplifying assumptions or closed numerical models, which simplify the flow, leading to inaccurate flow predictions or long solve times. This study examines solver constraints in a PINNs solver, aiming to generate an understanding of an optimal PINNs solver with reduced constraints compared with the numerically closed models used in traditional computational fluid dynamics (CFD). PINNs were implemented in a periodic hill flow case and compared with a simple data-driven approach to neural network modelling to show the limitations of a data-driven model on a small dataset (as is common in engineering design). A standard full equation PINNs model with predicted first-order stress terms was compared against reduced-boundary models and reduced-order models, with different levels of assumptions made about the flow to monitor the effect on the flow field predictions. The results in all cases showed good agreement against direct numerical simulation (DNS) data, with only boundary conditions provided for training as in numerical modelling. The efficacy of reduced-order models was shown using a continuity only model to accurately predict the flow fields within 0.147 and 2.6 percentage errors for streamwise and transverse velocities, respectively, and a modified mixing length model was used to show the effect of poor assumptions on the model, including poor convergence at the flow boundaries, despite a reduced solve time compared with a numerically closed equation set. The results agree with contemporary literature, indicating that physics-informed neural networks are a significant improvement in solve time compared with a data-driven approach, with a novel proposition of numerically derived unclosed equation sets being a good representation of a turbulent system. In conclusion, it is shown that numerically unclosed systems can be efficiently solved using reduced-order equation sets, potentially leading to a reduced compute requirement compared with traditional solver methods.

Proposal for an Intelligent Methodology to Manage Energy in Buildings and Detect Anomalies as a Compass Towards Zero Energy Buildings (ZEB)

This work aims to advance the optimisation of the efficiency of thermal installations in buildings, contributing to the achievement of Zero Energy Buildings (ZEB) in the context of maintenance and operation. This is achieved through an innovative proposal that merges machine learning techniques with thermoeconomics to perform diagnoses in building thermal systems and identify cost overruns generated by intrinsic anomalies in the components and quantify their induced effects on the rest of the components. To date, the few contributions combining these techniques have been limited to industrial applications and cost calculation, without addressing their application to building thermal systems, both from a dynamic perspective and for maintenance purposes. Research using Physics-Informed Neural Networks, PINNs, in this area is even scarcer, which underlines the complexity of defining a suitable methodology. Thus, the proposal integrates PINNs with a thermoeconomic diagnosis based on characteristic curves, allowing the comparison of the current operating condition with an anomaly-free reference condition to assess the existence of anomalies and their effects. For this reason, reference models are generated for the first time with PINNs, which represents a break with the conventional maintenance approaches used by professionals in the sector. Therefore, this methodology incorporates techniques that require specialised knowledge in thermodynamic and informatics areas, which motivates the present work to be focused on the exhaustive description of the methodology and to highlight the importance of continuing to explore lines of research in this unexplored field.

Deep neural networks for endemic measles dynamics: Comparative analysis and integration with mechanistic models.

Measles is an important infectious disease system both for its burden on public health and as an opportunity for studying nonlinear spatio-temporal disease dynamics. Traditional mechanistic models often struggle to fully capture the complex nonlinear spatio-temporal dynamics inherent in measles outbreaks. In this paper, we first develop a high-dimensional feed-forward neural network model with spatial features (SFNN) to forecast endemic measles outbreaks and systematically compare its predictive power with that of a classical mechanistic model (TSIR). We illustrate the utility of our model using England and Wales measles data from 1944-1965. These data present multiple modeling challenges due to the interplay between metapopulations, seasonal trends, and nonlinear dynamics related to demographic changes. Our results show that while the TSIR model yields similarly performant short-term (1 to 2 biweeks ahead) forecasts for highly populous cities, our neural network model (SFNN) consistently achieves lower root mean squared error (RMSE) across other forecasting windows. Furthermore, we show that our spatial-feature neural network model, without imposing mechanistic assumptions a priori, can uncover gravity-model-like spatial hierarchy of measles spread in which major cities play an important role in driving regional outbreaks. We then turn our attention to integrative approaches that combine mechanistic and machine learning models. Specifically, we investigate how the TSIR can be utilized to improve a state-of-the-art approach known as Physics-Informed-Neural-Networks (PINN) which explicitly combines compartmental models and neural networks. Our results show that the TSIR can facilitate the reconstruction of latent susceptible dynamics, thereby enhancing both forecasts in terms of mean absolute error (MAE) and parameter inference of measles dynamics within the PINN. In summary, our results show that appropriately designed neural network-based models can outperform traditional mechanistic models for short to long-term forecasts, while simultaneously providing mechanistic interpretability. Our work also provides valuable insights into more effectively integrating machine learning models with mechanistic models to enhance public health responses to measles and similar infectious disease systems.

An improved water strider algorithm for solving the inverse Burgers Huxley equation

In this paper, we introduce an improved water strider algorithm designed to solve the inverse form of the Burgers-Huxley equation, a nonlinear partial differential equation. Additionally, we propose a physics-informed neural network to address the same inverse problem. To demonstrate the effectiveness of the new algorithm and conduct a comparative analysis, we compare the results obtained using the improved water strider algorithm against those derived from the original water strider algorithm, a genetic algorithm, and a physics-informed neural network with three hidden layers. Solving the inverse form of nonlinear partial differential equations is crucial in many scientific and engineering applications, as it allows us to infer unknown parameters or initial conditions from observed data. This process is often challenging due to the complexity and nonlinearity of the equations involved. Meta-heuristic algorithms and neural networks have proven to be effective tools in addressing these challenges. The numerical results affirm the efficiency of our proposed method in solving the inverse form of the Burgers-Huxley equation. The best results were obtained using the improved water strider algorithm and the physics-informed neural network with 10,000 iterations. With this iteration count, the mean absolute error of these algorithms is O(10-4)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(10^{-4})$$\\end{document}. Additionally, the improved water strider algorithm is nearly four times faster than the physics-informed neural network. All algorithms were executed on a computing system equipped with an Intel(R) Core(TM) i7-7500U processor and 12.00 GB of RAM, and were implemented in MATLAB.

A unified framework for the error analysis of physics-informed neural networks

Abstract We prove a priori and a posteriori error estimates for physics-informed neural networks (PINNs) for linear PDEs. We analyze elliptic equations in primal and mixed form, elasticity, parabolic, hyperbolic and Stokes equations, and a PDE constrained optimization problem. For the analysis, we propose an abstract framework in the common language of bilinear forms, and we show that coercivity and continuity lead to error estimates. The obtained estimates are sharp and reveal that the $L^{2}$ penalty approach for initial and boundary conditions in the PINN formulation weakens the norm of the error decay. Finally, utilizing recent advances in PINN optimization, we present numerical examples that illustrate the ability of the method to achieve accurate solutions.

Solution to the Two-Phase Flow in Heterogeneous Porous Media Based on Physics-Informed Neural Network

Solution to the Two-Phase Flow in Heterogeneous Porous Media Based on Physics-Informed Neural Network

Pontryagin Neural Networks for the Class of Optimal Control Problems With Integral Quadratic Cost

This work introduces Pontryagin Neural Networks (PoNNs), a specialised subset of Physics-Informed Neural Networks (PINNs) that aim to learn optimal control actions for optimal control problems (OCPs) characterised by integral quadratic cost functions. PoNNs employ the Pontryagin Minimum Principle (PMP) to establish necessary conditions for optimality, resulting in a two-point boundary value problem (TPBVP) that involves both state and costate variables within a system of ordinary differential equations (ODEs). By modelling the unknown solutions of the TPBVP with neural networks, PoNNs effectively learn the optimal control strategies. We also derive upper bounds on the generalisation error of PoNNs in solving these OCPs, taking into account the selection and quantity of training points along with the training error. To validate our theoretical analysis, we perform numerical experiments on benchmark linear and nonlinear OCPs. The results indicate that PoNNs can successfully learn open-loop control actions for the considered class of OCPs, outperforming the commercial software GPOPS-II in terms of both accuracy and computational efficiency. The reduced computational time suggests that PoNNs hold promise for real-time applications.

Quantitative Analysis of the Fractional Fokker–Planck–Levy Equation via a Modified Physics-Informed Neural Network Architecture

An innovative approach is utilized in this paper to solve the fractional Fokker–Planck–Levy (FFPL) equation. A hybrid technique is designed by combining the finite difference method (FDM), Adams numerical technique, and physics-informed neural network (PINN) architecture, namely, the FDM-APINN, to solve the fractional Fokker–Planck–Levy (FFPL) equation numerically. Two scenarios of the FFPL equation are considered by varying the value of (i.e., 1.75, 1.85). Moreover, three cases of each scenario are numerically studied for different discretized domains with 100, 200, and 500 points in x∈[−1, 1] and t∈[0, 1]. For the FFPL equation, solutions are obtained via the FDM-APINN technique via 1000, 2000, and 5000 iterations. The errors, loss function graphs, and statistical tables are presented to validate our claim that the FDM-APINN is a better alternative intelligent technique for handling fractional-order partial differential equations with complex terms. The FDM-APINN can be extended by using nongradient-based bioinspired computing for higher-order fractional partial differential equations.

Physics-Informed Neural Networks for Modeling Li-ion Batteries: Solving the Single Particle Model Without Labeled Data

Abstract Li-ion batteries are garnering significant attention due to the electrification of critical sectors. High-fidelity battery cell models have proven effective in assessing performance and optimizing design, alleviating the financial burden associated with extensive experimental procedures. However, the computational costs associated with such simulations can become prohibitive, particularly when numerous iterations are required or when integration into small devices, such as battery management systems, is necessary. To address these challenges and provide an alternative to traditional methods such as finite element and finite volume solvers, we propose the development of an algorithm that utilizes Physics-Informed Neural Networks (PINNs) to solve the Single Particle Model across multiple parameter ranges. A notable advantage of this machine learning approach is its capacity to generate competitive solutions post-training by relying solely on the governing equations, without the necessity for experimental or simulation data. Additionally, the lightweight nature of the model indicates its potential for embedding within small-scale devices.

Chebyshev spectral approximation-based physics-informed neural network for solving higher-order nonlinear differential equations

Chebyshev spectral approximation-based physics-informed neural network for solving higher-order nonlinear differential equations

A brief review of reduced order models using intrusive and non‐intrusive techniques

AbstractReduced Order Models (ROMs) have gained a great attention by the scientific community in the last years thanks to their capabilities of significantly reducing the computational cost of the numerical simulations, which is a crucial objective in applications like real time control and shape optimization. This contribution aims to provide a brief overview about such a topic. We discuss both a classic intrusive framework based on a Galerkin projection technique and hybrid/non‐intrusive approaches, including Physics Informed Neural Networks (PINN), purely Data‐Driven Neural Networks (NN), Radial Basis Functions (RBF), Dynamic Mode Decomposition (DMD) and Gaussian Process Regression (GPR). We also briefly mention geometrical parametrization and dimensionality reduction methods like Active Subspaces (ASs). Then we test the performance of such approaches in terms of efficiency and accuracy against three academic test cases, the lid driven cavity, the flow past a cylinder and the geometrically parametrized Stanford Bunny. Moreover, we also briefly present some preliminary results related to a more complex case involving an industrial application.

Riemann-hilbert problem and physics-informed neural networks method for the nonlocal Sasa-Satsuma equation

Riemann-hilbert problem and physics-informed neural networks method for the nonlocal Sasa-Satsuma equation

A Physics Informed Neural Network for Solving the Inverse Heat Transfer Problem in Gas Turbine Rotating Cavities

Abstract Recently, Physics-Informed Neural Networks have displayed great potential in delivering swift and accurate solutions for inverse problems. Here, a Physics-Informed Neural Network (PINN) was used to solve the inverse heat conduction problem in the rotating cavities of a aeroengine high-pressure compressor internal air system. The neural network was designed to receive experimentally captured radially distributed temperature profiles as inputs and predict the associated surface heat fluxes. The correctness of these predicted heat fluxes are assessed by numerically solving the direct heat conduction equation using a 2D Finite-Element model, thereby recovering the original temperature profiles. A comparative analysis is conducted between the predicted temperature profiles and the initial inputs. The physics informed neural network is trained using noise free synthetic data, created from a range of radial temperature curve fit coefficients, and subsequently tested on noisy experimental data at engine representative conditions. The predicted temperature values exhibit some good agreement with their respective actual counterparts. Furthermore, the sensitivity of model hyperparameters are explored to showcase the capability of the proposed approach. The results show that Physics- Informed Neural Networks exhibit reduced susceptibility to experimental uncertainties when addressing inverse problems, in contrast to inverse solution methods, and offer a possible new approach for analysis of experimental data if trained on a sufficiently large dataset.

Computationally Efficient Inference via Time-Aware Modular Control Systems

Control in multi-agent decision-making systems is an important issue with a wide variety of existing approaches. In this work, we offer a new comprehensive framework for distributed control. The main contributions of this paper are summarized as follows. First, we propose PHIMEC (physics-informed meta control)—an architecture for learning optimal control by employing a physics-informed neural network when the state space is too large for reward-based learning. Second, we offer a way to leverage impulse response as a tool for system modeling and control. We propose IMPULSTM, a novel approach for incorporating time awareness into recurrent neural networks designed to accommodate irregular sampling rates in the signal. Third, we propose DIMAS, a modular approach to increasing computational efficiency in distributed control systems via domain-knowledge integration. We analyze the performance of the first two contributions on a set of corresponding benchmarks and then showcase their combined performance as a domain-informed distributed control system. The proposed approaches show satisfactory performance both individually in their respective applications and as a connected system.

Research on temperature prediction model of molten steel of tundish in continuous casting

To achieve the desired superheat of molten steel during the continuous casting process, optimization of process parameters such as molten steel temperature in ladle furnace, casting speed, and baking temperature is necessary. Therefore, obtaining the superheat corresponding to these process parameters in advance is particularly important. To address this issue, a model for predicting the temperature of molten steel in the tundish during continuous casting is designed. The model adopts a combined modeling approach of mechanistic model and data model. To address the issue of the mechanism model’s inability to capture the variation of the lining’s thermal parameters, this article improves the traditional physics-informed neural network (PINN) algorithm. It combines the constraints from both the forward and inverse problems, allowing for obtaining solutions to the equations while capturing the variation of equation parameters. Actual data from multiple casting sequences at a steel plant are collected to validate the accuracy and interpretability of the model. The results show that the error of the model is about 2.1k which has better accuracy compared to pure mechanistic model and pure data model. Additionally, it can capture the variation patterns of tundish lining thermal parameters under different operating conditions. Therefore, the model designed in this article can provide both profound physical interpretation ability and more practical predictions of molten steel temperature.