We introduce an optimized physics-informed neural network (PINN) trained to solve the problem of identifying and characterizing a surface breaking crack in a metal plate. PINNs are neural networks ...that can combine data and physics in the learning process by adding the residuals of a system of partial differential equations to the loss function. Our PINNs is supervised with realistic ultrasonic surface acoustic wave data acquired at a frequency of 5 MHz. The ultrasonic surface wave data is represented as a deformation on the top surface of a metal plate, measured by using the method of laser vibrometry. The PINN is physically informed by the acoustic wave equation and its convergence is sped up using adaptive activation functions. The adaptive activation function uses a trainable hyperparameter, which is optimized to achieve the best performance of the network. The adaptive activation function changes dynamically, involved in the optimization process. The usage of the adaptive activation function significantly improves the convergence, evidently observed in the current study. We use PINNs to estimate the speed of sound of the metal plate, which we do with an error of 1%, and then, by allowing the speed of sound to be space dependent, we identify and characterize the crack as the positions where the speed of sound has decreased. Our study also shows the effect of sub-sampling of the data on the sensitivity of sound speed estimates. More broadly, the resulting model shows a promising deep neural network model for ill-posed inverse problems.
Abstract
Phase-field modeling is an effective but computationally expensive method for capturing the mesoscale morphological and microstructure evolution in materials. Hence, fast and generalizable ...surrogate models are needed to alleviate the cost of computationally taxing processes such as in optimization and design of materials. The intrinsic discontinuous nature of the physical phenomena incurred by the presence of sharp phase boundaries makes the training of the surrogate model cumbersome. We develop a framework that integrates a convolutional autoencoder architecture with a deep neural operator (DeepONet) to learn the dynamic evolution of a two-phase mixture and accelerate time-to-solution in predicting the microstructure evolution. We utilize the convolutional autoencoder to provide a compact representation of the microstructure data in a low-dimensional latent space. After DeepONet is trained in the latent space, it can be used to replace the high-fidelity phase-field numerical solver in interpolation tasks or to accelerate the numerical solver in extrapolation tasks.
Discovering mathematical equations that govern physical and biological systems from observed data is a fundamental challenge in scientific research. We present a new physics-informed framework for ...parameter estimation and missing physics identification (gray-box) in the field of Systems Biology. The proposed framework-named AI-Aristotle-combines the eXtreme Theory of Functional Connections (X-TFC) domain-decomposition and Physics-Informed Neural Networks (PINNs) with symbolic regression (SR) techniques for parameter discovery and gray-box identification. We test the accuracy, speed, flexibility, and robustness of AI-Aristotle based on two benchmark problems in Systems Biology: a pharmacokinetics drug absorption model and an ultradian endocrine model for glucose-insulin interactions. We compare the two machine learning methods (X-TFC and PINNs), and moreover, we employ two different symbolic regression techniques to cross-verify our results. To test the performance of AI-Aristotle, we use sparse synthetic data perturbed by uniformly distributed noise. More broadly, our work provides insights into the accuracy, cost, scalability, and robustness of integrating neural networks with symbolic regressors, offering a comprehensive guide for researchers tackling gray-box identification challenges in complex dynamical systems in biomedicine and beyond.
Celotno besedilo
Dostopno za:
DOBA, IZUM, KILJ, NUK, PILJ, PNG, SAZU, SIK, UILJ, UKNU, UL, UM, UPUK
We use machine learning algorithms (artificial neural networks, ANNs) to estimate petrophysical models at seismic scale combining well-log information, seismic data and seismic attributes. The ...resulting petrophysical images are the prior inputs in the process of full-waveform inversion (FWI). We calculate seismic attributes from a stacked reflected 2-D seismic section and then train ANNs to approximate the following petrophysical parameters: P-wave velocity (
V
p
), density (
ρ
) and volume of clay (
V
clay
). We extend the use of the
V
clay
by constraining it with the well lithology and we establish two classes: sands and shales. Consequently, machine learning allows us to build an initial estimate of the earth property model (
V
p
), which is iteratively refined to produce a synthetic seismogram that matches the observed seismic data. We apply the 1-D Kennett method as a forward modeling tool to create synthetic data with the images of
V
p
,
ρ
and the thickness of layers (sands or shales) obtained with the ANNs. A nonlinear least-squares inversion algorithm minimizes the residual (or misfit) between observed and synthetic full-waveform data, which improves the
V
p
resolution. In order to show the advantage of using the ANN velocity model as the initial velocity model for the inversion, we compare the results obtained with the ANNs and two other initial velocity models. One of these alternative initial velocity models is computed via P-wave impedance, and the other is achieved by velocity semblance analysis: root-mean-square velocity (RMS). The results are in good agreement when we use
ρ
and
V
p
obtained by ANNs. However, the results are poor and the synthetic data do not match the real acquired data when using the semblance velocity model and the
ρ
from the well log (constant for the entire 2-D section). Nevertheless, the results improve when including
ρ
, the layered structure driven by the
V
clay
(both obtained with ANNs) and the semblance velocity model. When doing inversion starting with the initial
V
p
model estimated using the P-wave impedance, there is some gain of the final
V
p
with respect to the RMS initial
V
p
. To assess the quality of the inversion of
V
p
, we use the information for two available wells and compare the final
V
p
obtained with ANNs and the final
V
p
computed with the P-wave impedance. This shows the benefit of employing ANNs estimations as prior models during the inversion process to obtain a final
V
p
that is in agreement with the geology and with the seismic and well-log data. To illustrate the computation of the final velocity model via FWI, we provide an algorithm with the detailed steps and its corresponding GitHub code.
Abstract
Physics-informed machine learning (PIML) has emerged as a promising new approach for simulating complex physical and biological systems that are governed by complex multiscale processes for ...which some data are also available. In some instances, the objective is to discover part of the hidden physics from the available data, and PIML has been shown to be particularly effective for such problems for which conventional methods may fail. Unlike commercial machine learning where training of deep neural networks requires big data, in PIML big data are not available. Instead, we can train such networks from additional information obtained by employing the physical laws and evaluating them at random points in the space–time domain. Such PIML integrates multimodality and multifidelity data with mathematical models, and implements them using neural networks or graph networks. Here, we review some of the prevailing trends in embedding physics into machine learning, using physics-informed neural networks (PINNs) based primarily on feed-forward neural networks and automatic differentiation. For more complex systems or systems of systems and unstructured data, graph neural networks (GNNs) present some distinct advantages, and here we review how physics-informed learning can be accomplished with GNNs based on graph exterior calculus to construct differential operators; we refer to these architectures as physics-informed graph networks (PIGNs). We present representative examples for both forward and inverse problems and discuss what advances are needed to scale up PINNs, PIGNs and more broadly GNNs for large-scale engineering problems.
•Construction and implementation of new domain-decomposition based parallel algorithm is proposed for cPINNs and XPINNs methods.•The proposed algorithm adds another dimension of parallelism in SciML ...primarily driven by data and model parallelism.•The proposed algorithm is shown its scaling for CPU and CPU+GPU architectures.
We develop a distributed framework for the physics-informed neural networks (PINNs) based on two recent extensions, namely conservative PINNs (cPINNs) and extended PINNs (XPINNs), which employ domain decomposition in space and in time-space, respectively. This domain decomposition endows cPINNs and XPINNs with several advantages over the vanilla PINNs, such as parallelization capacity, large representation capacity, efficient hyperparameter tuning, and is particularly effective for multi-scale and multi-physics problems. Here, we present a parallel algorithm for cPINNs and XPINNs constructed with a hybrid programming model described by MPI + X, where X ∈{CPUs,GPUs}. The main advantage of cPINN and XPINN over the more classical data and model parallel approaches is the flexibility of optimizing all hyperparameters of each neural network separately in each subdomain. We compare the performance of distributed cPINNs and XPINNs for various forward problems, using both weak and strong scalings. Our results indicate that for space domain decomposition, cPINNs are more efficient in terms of communication cost but XPINNs provide greater flexibility as they can also handle time-domain decomposition for any differential equations, and can deal with any arbitrarily shaped complex subdomains. To this end, we also present an application of the parallel XPINN method for solving an inverse diffusion problem with variable conductivity on the United States map, using ten regions as subdomains.
We compare high-order methods including spectral difference (SD), flux reconstruction (FR), the entropy-stable discontinuous Galerkin spectral element method (ES-DGSEM), modal discontinuous Galerkin ...methods, and WENO to select the best candidate to simulate strong shock waves characteristic of hypersonic flows. We consider several benchmarks, including the Leblanc and modified shock-density wave interaction problems that require robust stabilization and positivity-preserving properties for a successful flow realization. We also perform simulations of the three-species Sod problem with simplified chemistry with the chemical reaction source terms introduced in the Euler equations. The ES-DGSEM scheme exhibits the highest stability, negligible numerical oscillations, and requires the least computational effort in resolving reactive flow regimes with strong shock waves. Therefore, we extend the ES-DGSEM to hypersonic Euler equations by deriving a new set of two-point entropy conservative fluxes for a five-species gas model. In this paper, hypersonic Euler equations refer to the multi-species Euler equations for which the internal energy and thermodynamic properties are computed using the Rigid-Rotor Harmonic-Oscillator model. Stabilization for capturing strong shock waves occurs by blending high-order entropy conservative fluxes with low-order finite volume fluxes constructed using the HLLC Riemann solver. The hypersonic Euler solver is verified using the non-equilibrium chemistry Sod problem. To this end, we adopt the Mutation++ library to compute the reaction source terms, thermodynamic properties, and transport coefficients. We also investigate the effect of real chemistry versus ideal chemistry, and the results demonstrate that the ideal chemistry assumption fails at high temperatures, hence real chemistry must be employed for accurate predictions. Finally, we consider a viscous hypersonic flow problem to verify the transport coefficients and reaction source terms determined by the Mutation++ library.
•Resolving extreme pressure ratio shock waves in near vacuum conditions.•Constructing entropy-stable high order schemes for the hypersonic Euler equations.•Deriving Entropy-conservative fluxes for flows modeled with the RRHO method.•Comparing real chemistry versus ideal chemistry in hypersonic flows.•Hypersonic NS solver based on the entropy-stable Euler scheme and the BR1 approach.
High order schemes are known to be unstable in the presence of shock discontinuities or under-resolved solution features for nonlinear conservation laws. Entropy stable schemes address this ...instability by ensuring that physically relevant solutions satisfy a semi-discrete entropy inequality independently of discretization parameters. This work extends high order entropy stable schemes to the quasi-1D shallow water equations and the quasi-1D compressible Euler equations, which model one-dimensional flows through channels or nozzles with varying width.
We introduce new non-symmetric entropy conservative finite volume fluxes for both sets of quasi-1D equations, as well as a generalization of the entropy conservation condition to non-symmetric fluxes. When combined with an entropy stable interface flux, the resulting schemes are high order accurate, conservative, and semi-discretely entropy stable. For the quasi-1D shallow water equations, the resulting schemes are also well-balanced.
•Extend entropy stable high order summation-by-parts schemes to the quasi-1D shallow water and compressible Euler equations.•We introduce a non-symmetric “Tadmor shuffle” condition to simplify the analysis of entropy conservation.•We validate the discretization for several problems and channel geometries.
The curse-of-dimensionality taxes computational resources heavily with exponentially increasing computational cost as the dimension increases. This poses great challenges in solving high-dimensional ...partial differential equations (PDEs), as Richard E. Bellman first pointed out over 60 years ago. While there has been some recent success in solving numerical PDEs in high dimensions, such computations are prohibitively expensive, and true scaling of general nonlinear PDEs to high dimensions has never been achieved. We develop a new method of scaling up physics-informed neural networks (PINNs) to solve arbitrary high-dimensional PDEs. The new method, called Stochastic Dimension Gradient Descent (SDGD), decomposes a gradient of PDEs’ and PINNs’ residual into pieces corresponding to different dimensions and randomly samples a subset of these dimensional pieces in each iteration of training PINNs. We prove theoretically the convergence and other desired properties of the proposed method. We demonstrate in various diverse tests that the proposed method can solve many notoriously hard high-dimensional PDEs, including the Hamilton–Jacobi-Bellman (HJB) and the Schrödinger equations in tens of thousands of dimensions very fast on a single GPU using the PINNs mesh-free approach. Notably, we solve nonlinear PDEs with nontrivial, anisotropic, and inseparable solutions in less than one hour for 1000 dimensions and in 12 h for 100,000 dimensions on a single GPU using SDGD with PINNs. Since SDGD is a general training methodology of PINNs, it can be applied to any current and future variants of PINNs to scale them up for arbitrary high-dimensional PDEs.
We propose a framework and an algorithm to uncover the unknown parts of nonlinear equations directly from data. The framework is based on eXtended Physics-Informed Neural Networks (X-PINNs), domain ...decomposition in space–time, but we augment the original X-PINN method by imposing flux continuity across the domain interfaces. The well-known Allen–Cahn equation is used to demonstrate the approach. The Frobenius matrix norm is used to evaluate the accuracy of the X-PINN predictions and the results show excellent performance. In addition, symbolic regression is employed to determine the closed form of the unknown part of the equation from the data, and the results confirm the accuracy of the X-PINNs based approach. To test the framework in a situation resembling real-world data, random noise is added to the datasets to mimic scenarios such as the presence of thermal noise or instrument errors. The results show that the framework is stable against significant amount of noise. As the final part, we determine the minimal amount of data required for training the neural network. For the systems studied here, the framework is able to predict the correct form of the underlying dynamical equation when at least 50% data is used for training. However, relying solely on 50% of the data is inadequate for accurately predicting the unknown coefficients. To ensure accurate predictions for the coefficients, it was necessary to train the network with a minimum of 60% of the available data.
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