Despite great progress in simulating multiphysics problems using the numerical discretization of partial differential equations (PDEs), one still cannot seamlessly incorporate noisy data into existing algorithms, mesh generation remains complex, and high-dimensional problems governed by parameterized PDEs cannot be tackled. Moreover, solving inverse problems with hidden physics is often prohibitively expensive and requires different formulations and elaborate computer codes. Machine learning has emerged as a promising alternative, but training deep neural networks requires big data, not always available for scientific problems. Instead, such networks can be trained from additional information obtained by enforcing the physical laws (for example, at random points in the continuous space-time domain). Such physics-informed learning integrates (noisy) data and mathematical models, and implements them through neural networks or other kernel-based regression networks. Moreover, it may be possible to design specialized network architectures that automatically satisfy some of the physical invariants for better accuracy, faster training and improved generalization. Here, we review some of the prevailing trends in embedding physics into machine learning, present some of the current capabilities and limitations and discuss diverse applications of physics-informed learning both for forward and inverse problems, including discovering hidden physics and tackling high-dimensional problems. The rapidly developing field of physics-informed learning integrates data and mathematical models seamlessly, enabling accurate inference of realistic and high-dimensional multiphysics problems. This Review discusses the methodology and provides diverse examples and an outlook for further developments.

Physics-informed machine learning

A physics-informed deep learning framework for inversion and surrogate modeling in solid mechanics

\n Physics-informed neural networks (PINNs) have gained popularity across different engineering fields due to their effectiveness in solving realistic problems with noisy data and often partially missing physics. In PINNs, automatic differentiation is leveraged to evaluate differential operators without discretization errors, and a multitask learning problem is defined in order to simultaneously fit observed data while respecting the underlying governing laws of physics. Here, we present applications of PINNs to various prototype heat transfer problems, targeting in particular realistic conditions not readily tackled with traditional computational methods. To this end, we first consider forced and mixed convection with unknown thermal boundary conditions on the heated surfaces and aim to obtain the temperature and velocity fields everywhere in the domain, including the boundaries, given some sparse temperature measurements. We also consider the prototype Stefan problem for two-phase flow, aiming to infer the moving interface, the velocity and temperature fields everywhere as well as the different conductivities of a solid and a liquid phase, given a few temperature measurements inside the domain. Finally, we present some realistic industrial applications related to power electronics to highlight the practicality of PINNs as well as the effective use of neural networks in solving general heat transfer problems of industrial complexity. Taken together, the results presented herein demonstrate that PINNs not only can solve ill-posed problems, which are beyond the reach of traditional computational methods, but they can also bridge the gap between computational and experimental heat transfer.

Physics-Informed Neural Networks for Heat Transfer Problems

Abstract Physics-Informed Neural Networks (PINN) are neural networks (NNs) that encode model equations, like Partial Differential Equations (PDE), as a component of the neural network itself. PINNs are nowadays used to solve PDEs, fractional equations, integral-differential equations, and stochastic PDEs. This novel methodology has arisen as a multi-task learning framework in which a NN must fit observed data while reducing a PDE residual. This article provides a comprehensive review of the literature on PINNs: while the primary goal of the study was to characterize these networks and their related advantages and disadvantages. The review also attempts to incorporate publications on a broader range of collocation-based physics informed neural networks, which stars form the vanilla PINN, as well as many other variants, such as physics-constrained neural networks (PCNN), variational hp-VPINN, and conservative PINN (CPINN). The study indicates that most research has focused on customizing the PINN through different activation functions, gradient optimization techniques, neural network structures, and loss function structures. Despite the wide range of applications for which PINNs have been used, by demonstrating their ability to be more feasible in some contexts than classical numerical techniques like Finite Element Method (FEM), advancements are still possible, most notably theoretical issues that remain unresolved. 

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Scientific Machine Learning Through Physics–Informed Neural Networks: Where we are and What’s Next

Physics-Informed Neural Networks (PINNs) have emerged recently as a promising application of deep neural networks to the numerical solution of nonlinear partial differential equations (PDEs). However, the original PINN algorithm is known to suffer from stability and accuracy problems in cases where the solution has sharp spatio-temporal transitions. These stiff PDEs require an unreasonably large number of collocation points to be solved accurately. It has been recognized that adaptive procedures are needed to force the neural network to fit accurately the stubborn spots in the solution of stiff PDEs. To accomplish this, previous approaches have used fixed weights hard-coded over regions of the solution deemed to be important. In this paper, we propose a fundamentally new method to train PINNs adaptively, where the adaptation weights are fully trainable, so the neural network learns by itself which regions of the solution are difficult and is forced to focus on them, which is reminiscent of soft multiplicative-mask attention mechanism used in computer vision. The basic idea behind these Self-Adaptive PINNs is to make the weights increase where the corresponding loss is higher, which is accomplished by training the network to simultaneously minimize the losses and maximize the weights, i.e., to find a saddle point in the cost surface. We show that this is formally equivalent to solving a PDE-constrained optimization problem using a penalty-based method, though in a way where the monotonically-nondecreasing penalty coefficients are trainable. Numerical experiments with an Allen-Cahn stiff PDE, the Self-Adaptive PINN outperformed other state-of-the-art PINN algorithms in L2 error by a wide margin, while using a smaller number of training epochs. An Appendix contains additional results with Burger's and Helmholtz PDEs, which confirmed the trends observed in the Allen-Cahn experiments.

Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanism

Physics-informed neural networks (PINNs) have lately received great attention thanks to their flexibility in tackling a wide range of forward and inverse problems involving partial differential equations. However, despite their noticeable empirical success, little is known about how such constrained neural networks behave during their training via gradient descent. More importantly, even less is known about why such models sometimes fail to train at all. In this work, we aim to investigate these questions through the lens of the Neural Tangent Kernel (NTK); a kernel that captures the behavior of fully-connected neural networks in the infinite width limit during training via gradient descent. Specifically, we derive the NTK of PINNs and prove that, under appropriate conditions, it converges to a deterministic kernel that stays constant during training in the infinite-width limit. This allows us to analyze the training dynamics of PINNs through the lens of their limiting NTK and find a remarkable discrepancy in the convergence rate of the different loss components contributing to the total training error. To address this fundamental pathology, we propose a novel gradient descent algorithm that utilizes the eigenvalues of the NTK to adaptively calibrate the convergence rate of the total training error. Finally, we perform a series of numerical experiments to verify the correctness of our theory and the practical effectiveness of the proposed algorithms. The data and code accompanying this manuscript are publicly available at \url{this https URL}.

When and why PINNs fail to train: A neural tangent kernel perspective

Physics-informed neural networks (PINNs) have been increasingly employed due to their capability of modeling complex physics systems. To achieve better expressiveness, increasingly larger network sizes are required in many problems. This has caused challenges when we need to train PINNs on edge devices with limited memory, computing and energy resources. To enable training PINNs on edge devices, this paper proposes an end-to-end compressed PINN based on Tensor-Train decomposition. In solving a Helmholtz equation, our proposed model significantly outperforms the original PINNs with few parameters and achieves satisfactory prediction with up to 15$\times$ overall parameter reduction.

TT-PINN: A Tensor-Compressed Neural PDE Solver for Edge Computing

Electrical properties (EP), namely permittivity and electric conductivity, dictate the interactions between electromagnetic waves and biological tissue. EP can be potential biomarkers for pathology characterization, such as cancer, and improve therapeutic modalities, such radiofrequency hyperthermia and ablation. MR-based electrical properties tomography (MR-EPT) uses MR measurements to reconstruct the EP maps. Using the homogeneous Helmholtz equation, EP can be directly computed through calculations of second order spatial derivatives of the measured magnetic transmit or receive fields $(B_{1}^{+}, B_{1}^{-})$. However, the numerical approximation of derivatives leads to noise amplifications in the measurements and thus erroneous reconstructions. Recently, a noise-robust supervised learning-based method (DL-EPT) was introduced for EP reconstruction. However, the pattern-matching nature of such network does not allow it to generalize for new samples since the network's training is done on a limited number of simulated data. In this work, we leverage recent developments on physics-informed deep learning to solve the Helmholtz equation for the EP reconstruction. We develop deep neural network (NN) algorithms that are constrained by the Helmholtz equation to effectively de-noise the $B_{1}^{+}$ measurements and reconstruct EP directly at an arbitrarily high spatial resolution without requiring any known $B_{1}^{+}$ and EP distribution pairs.

MR-Based Electrical Property Reconstruction Using Physics-Informed Neural Networks

\textit{Objective:} In this paper, we introduce Physics-Informed Fourier Networks for Electrical Properties Tomography (PIFON-EPT), a novel deep learning-based method that solves an inverse scattering problem based on noisy and/or incomplete magnetic resonance (MR) measurements. \textit{Methods:} We used two separate fully-connected neural networks, namely $B_1^{+}$ Net and EP Net, to solve the Helmholtz equation in order to learn a de-noised version of the input $B_1^{+}$ maps and estimate the object's EP. A random Fourier features mapping was embedded into $B_1^{+}$ Net, to learn the high-frequency details of $B_1^{+}$ more efficiently. The two neural networks were trained jointly by minimizing the combination of a physics-informed loss and a data mismatch loss via gradient descent. \textit{Results:} We performed several numerical experiments, showing that PIFON-EPT could provide physically consistent reconstructions of the EP and transmit field. Even when only $50\%$ of the noisy MR measurements were used as inputs, our method could still reconstruct the EP and transmit field with average error $2.49\%$, $4.09\%$ and $0.32\%$ for the relative permittivity, conductivity and $B_{1}^{+}$, respectively, over the entire volume of the phantom. The generalized version of PIFON-EPT that accounts for gradients of EP yielded accurate results at the interface between regions of different EP values without requiring any boundary conditions. \textit{Conclusion:} This work demonstrated the feasibility of PIFON-EPT, suggesting it could be an accurate and effective method for EP estimation. \textit{Significance:} PIFON-EPT can efficiently de-noise $B_1^{+}$ maps, which has the potential to improve other MR-based EPT techniques. Furthermore, PIFON-EPT is the first technique that can reconstruct EP and $B_{1}^{+}$ simultaneously from incomplete noisy MR measurements.

Xinling Yu

Papers

When and why PINNs fail to train: A neural tangent kernel perspective

When and why PINNs fail to train: A neural tangent kernel perspective

TT-PINN: A Tensor-Compressed Neural PDE Solver for Edge Computing

MR-Based Electrical Property Reconstruction Using Physics-Informed Neural Networks

PIFON-EPT: MR-Based Electrical Property Tomography Using Physics-Informed Fourier Networks