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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

Tackling the curse of dimensionality in fractional and tempered fractional PDEs with physics-informed neural networks

Fractional and tempered fractional partial differential equations (PDEs) are effective models of long-range interactions, anomalous diffusion, and non-local effects. Traditional numerical methods for these problems are mesh-based, thus struggling with the curse of dimensionality (CoD). Physics-informed neural networks (PINNs) offer a promising solution due to their universal approximation, generalization ability, and mesh-free training. In principle, Monte Carlo fractional PINN (MC-fPINN) estimates fractional derivatives using Monte Carlo methods and thus could lift CoD. However, this may cause significant variance and errors, hence affecting convergence; in addition, MC-fPINN is sensitive to hyperparameters. In general, numerical methods and specifically PINNs for tempered fractional PDEs are under-developed. Herein, we extend MC-fPINN to tempered fractional PDEs to address these issues, resulting in the Monte Carlo tempered fractional PINN (MC-tfPINN). To reduce possible high variance and errors from Monte Carlo sampling, we replace the one-dimensional (1D) Monte Carlo with 1D Gaussian quadrature, applicable to both MC-fPINN and MC-tfPINN. We validate our methods on various forward and inverse problems of fractional and tempered fractional PDEs, scaling up to 100,000 dimensions. Our improved MC-fPINN/MC-tfPINN using quadrature consistently outperforms the original versions in accuracy and convergence speed in very high dimensions.

42 ENGINEERING↗
DOE OSTI2020

Extended Physics-Informed Neural Networks (XPINNs): A Generalized Space-Time Domain Decomposition Based Deep Learning Framework for Nonlinear Partial Differential Equations

Here we propose a generalized space-time domain decomposition approach for the physics-informed neural networks (PINNs) to solve nonlinear partial differential equations (PDEs) on arbitrary complex-geometry domains. The proposed framework, named eXtended PINNs ( X P I N N s ), further pushes the boundaries of both PINNs as well as conservative PINNs (cPINNs), which is a recently proposed domain decomposition approach in the PINN framework tailored to conservation laws. Compared to PINN, the XPINN method has large representation and parallelization capacity due to the inherent property of deployment of multiple neural networks in the smaller subdomains. Unlike cPINN, XPINN can be extended to any type of PDEs. Moreover, the domain can be decomposed in any arbitrary way (in space and time), which is not possible in cPINN. Thus, XPINN offers both space and time parallelization, thereby reducing the training cost more effectively. In each subdomain, a separate neural network is employed with optimally selected hyperparameters, e.g., depth/width of the network, number and location of residual points, activation function, optimization method, etc. A deep network can be employed in a subdomain with complex solution, whereas a shallow neural network can be used in a subdomain with relatively simple and smooth solutions. We demonstrate the versatility of XPINN by solving both forward and inverse PDE problems, ranging from one-dimensional to three-dimensional problems, from time-dependent to time-independent problems, and from continuous to discontinuous problems, which clearly shows that the XPINN method is promising in many practical problems. The proposed XPINN method is the generalization of PINN and cPINN methods, both in terms of applicability as well as domain decomposition approach, which efficiently lends itself to parallelized computation. The XPINN code is available on h t t p s : / / g i t h u b . c o m / A m e y a J a g t a p / X P I N N s .

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Differentiable Earth mover’s distance for data compression at the high-luminosity LHC

Abstract The Earth mover’s distance (EMD) is a useful metric for image recognition and classification, but its usual implementations are not differentiable or too slow to be used as a loss function for training other algorithms via gradient descent. In this paper, we train a convolutional neural network (CNN) to learn a differentiable, fast approximation of the EMD and demonstrate that it can be used as a substitute for computing-intensive EMD implementations. We apply this differentiable approximation in the training of an autoencoder-inspired neural network (encoder NN) for data compression at the high-luminosity LHC at CERN The goal of this encoder NN is to compress the data while preserving the information related to the distribution of energy deposits in particle detectors. We demonstrate that the performance of our encoder NN trained using the differentiable EMD CNN surpasses that of training with loss functions based on mean squared error.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
DOE OSTI2022

Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems

Deep learning has been shown to be an effective tool in solving partial differential equations (PDEs) through physics-informed neural networks (PINNs). PINNs embed the PDE residual into the loss function of the neural network, and have been successfully employed to solve diverse forward and inverse PDE problems. However, one disadvantage of the first generation of PINNs is that they usually have limited accuracy even with many training points. Here, we propose a new method, gradient-enhanced physics-informed neural networks (gPINNs), for improving the accuracy of PINNs. gPINNs leverage gradient information of the PDE residual and embed the gradient into the loss function. We tested gPINNs extensively and demonstrated the effectiveness of gPINNs in both forward and inverse PDE problems. Our numerical results show that gPINN performs better than PINN with fewer training points. Additionally, we combined gPINN with the method of residual-based adaptive refinement (RAR), a method for improving the distribution of training points adaptively during training, to further improve the performance of gPINN, especially in PDEs with solutions that have steep gradients.

42 ENGINEERING↗
DOE OSTI2021

NEURAL NETWORK FOR COHERENT DIFFRACTION IMAGE INVERSION

A deep neural network model plus automatic differentiation is developed for retrieving phase information from 3D coherent diffraction images. The model is implemented using Tensorflow and the training dataset is generated using physics-based atomistic simulations. Custom codes are written to handle the resampling of diffraction images to oversampling ratios appropriate for the neural network model.

CHAN, HENRY↗
DOE OSTI2022

A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks

Physics-informed neural networks (PINNs) have shown to be effective tools for solving both forward and inverse problems of partial differential equations (PDEs). PINNs embed the PDEs into the loss of the neural network using automatic differentiation, and this PDE loss is evaluated at a set of scattered spatio-temporal points (called residual points). The location and distribution of these residual points are highly important to the performance of PINNs. However, in the existing studies on PINNs, only a few simple residual point sampling methods have mainly been used. Here, we present a comprehensive study of two categories of sampling for PINNs: non-adaptive uniform sampling and adaptive nonuniform sampling. We consider six uniform sampling methods, including (1) equispaced uniform grid, (2) uniformly random sampling, (3) Latin hypercube sampling, (4) Halton sequence, (5) Hammersley sequence, and (6) Sobol sequence. We also consider a resampling strategy for uniform sampling. To improve the sampling efficiency and the accuracy of PINNs, we propose two new residual-based adaptive sampling methods: residual-based adaptive distribution (RAD) and residual-based adaptive refinement with distribution (RAR-D), which dynamically improve the distribution of residual points based on the PDE residuals during training. Hence, we have considered a total of 10 different sampling methods, including six non-adaptive uniform sampling, uniform sampling with resampling, two proposed adaptive sampling, and an existing adaptive sampling. We extensively tested the performance of these sampling methods for four forward problems and two inverse problems in many setups. Our numerical results presented in this study are summarized from more than 6000 simulations of PINNs. Here, we show that the proposed adaptive sampling methods of RAD and RAR-D significantly improve the accuracy of PINNs with fewer residual points for both forward and inverse problems. Furthermore, the results obtained in this study can also be used as a practical guideline in choosing sampling methods.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Reliable extrapolation of deep neural operators informed by physics or sparse observations

Deep neural operators can learn nonlinear mappings between infinite-dimensional function spaces via deep neural networks. As promising surrogate solvers of partial differential equations (PDEs) for real-time prediction, deep neural operators such as deep operator networks (DeepONets) provide a new simulation paradigm in science and engineering. Pure data-driven neural operators and deep learning models, in general, are usually limited to interpolation scenarios, where new predictions utilize inputs within the support of the training set. However, in the inference stage of real-world applications, the input may lie outside the support, i.e., extrapolation is required, which may result to large errors and unavoidable failure of deep learning models. Here, we address this challenge of extrapolation for deep neural operators. First, we systematically investigate the extrapolation behavior of DeepONets by quantifying the extrapolation complexity, via the 2-Wasserstein distance between two function spaces and propose a new strategy of bias–variance trade-off for extrapolation with respect to model capacity. Subsequently, we develop a complete workflow, including extrapolation determination, and we propose five reliable learning methods that guarantee a safe prediction under extrapolation by requiring additional information—the governing PDEs of the system or sparse new observations. The proposed methods are based on either fine-tuning a pre-trained DeepONet or multifidelity learning. We demonstrate the effectiveness of the proposed framework for various types of parametric PDEs. Furthermore, our systematic comparisons provide practical guidelines for selecting a proper extrapolation method depending on the available information, desired accuracy, and required inference speed.

42 ENGINEERING↗
DOE OSTI2022

Machine learning of consistent thermodynamic models using automatic differentiation

In this study, we propose a data-driven method to describe consistent equations of state (EOS) for arbitrary systems. Complex EOS are traditionally obtained by fitting suitable analytical expressions to thermophysical data. A key aspect of EOS is that the relationships between state variables are given by derivatives of the system free energy. In this work, we model the free energy with an artificial neural network and utilize automatic differentiation to directly learn the derivatives of the free energy. We demonstrate this approach on two different systems, the analytic van der Waals EOS and published data for the Lennard-Jones fluid, and we show that it is advantageous over direct learning of thermodynamic properties (i.e., not as derivatives of the free energy but as independent properties), in terms of both accuracy and the exact preservation of the Maxwell relations. Furthermore, the method implicitly provides the free energy of a system without explicit integration.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2022

Inverse Analysis with Variational Autoencoders: A Comparison of Shallow and Deep Networks

Inverse problems are applied to determine unknown properties by matching observational data with a physical model that often takes many parameters as input. To overcome the underconstrained nature of inverse problems and achieve good performance, an approach is presented involving regularization with a technique known as a variational autoencoder (VAE), which is trained to map a high-dimensional parameter space with a complex structure to a low-dimensional latent space with a simple structure. We apply this approach to unconditioned realizations of the parameters (heterogeneous hydraulic fields) for a hydrogeological inverse problem. Two types of hydraulic conductivity fields are used to evaluate the characterization for the different levels of heterogeneity complexity of the physical inputs. This approach keeps the computational cost of generating the training data low. The reason is unconditioned realizations neither rely on the observational data used to perform the inverse analysis nor require any groundwater flow model forward runs. In addition, this approach applies regularization on a low-dimensional latent space from the VAE and increases optimization efficiency through automatic differentiation. Furthermore, two different neural network (NN) structures are tested for their utility in using the VAE for inverse analysis. The performance of a deep, convolutional neural network strongly depends on the dimensionality of the latent space, which requires tuning. In contrast, a shallow, dense neural network provides consistently accurate characterization without tuning. Furthermore, our approach evaluates the advantages of the shallow, dense neural network over the deep, convolutional one and enables future application to a wide range of inverse problems.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

A differentiable approach to the maximum independent set problem using dataless neural networks

The success of machine learning solutions for reasoning about discrete structures has brought attention to its adoption within combinatorial optimization algorithms. Such approaches generally rely on supervised learning by leveraging datasets of the combinatorial structures of interest drawn from some distribution of problem instances. Reinforcement learning has also been employed to find such structures. Here, in this paper, we propose a different approach in that no data is required for training the neural networks that produce the solution. In this sense, what we present is not a machine learning solution, but rather one that is dependent on neural networks and where backpropagation is applied to a loss function defined by the structure of the neural network architecture as opposed to a training dataset. In particular, we reduce the popular combinatorial optimization problem of finding a maximum independent set to a neural network and employ a dataless training scheme to refine the parameters of the network such that those parameters yield the structure of interest. Additionally, we propose a universal graph reduction procedure to handle large-scale graphs. The reduction exploits community detection for graph partitioning and is applicable to any graph type and/or density. Experimental results on both real and synthetic graphs demonstrate that our proposed method performs on par or outperforms state-of-the-art learning-based methods in terms of the size of the found set without requiring any training data.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Time-series learning of latent-space dynamics for reduced-order model closure

In this work, we study the performance of long short-term memory networks (LSTMs) and neural ordinary differential equations (NODEs) in learning latent-space representations of dynamical equations for an advection-dominated problem given by the viscous Burgers equation. Our formulation is devised in a nonintrusive manner with an equation-free evolution of dynamics in a reduced space with the latter being obtained through a proper orthogonal decomposition. In addition, we leverage the sequential nature of learning for both LSTMs and NODEs to demonstrate their capability for closure in systems that are not completely resolved in the reduced space. We assess our hypothesis for two advection-dominated problems given by the viscous Burgers equation. We observe that both LSTMs and NODEs are able to reproduce the effects of the absent scales for our test cases more effectively than does intrusive dynamics evolution through a Galerkin projection. This result empirically suggests that time-series learning techniques implicitly leverage a memory kernel for coarse-grained system closure as is suggested through the Mori–Zwanzig formalism.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Adaptive Data-Driven Deep-Learning Surrogate Model for Frontal Polymerization in Dicyclopentadiene

Frontal polymerization (FP) is a self-sustaining curing process that enables rapid and energy-efficient manufacturing of thermoset polymers and composites. Computational methods conventionally used to simulate the FP process are time-consuming, and repeating simulations are required for sensitivity analysis, uncertainty quantification, or optimization of the manufacturing process. Here, in this work, we develop an adaptive surrogate deep-learning model for FP of dicyclopentadiene (DCPD), which predicts the evolution of temperature and degree of cure orders of magnitude faster than the finite-element method (FEM). The adaptive algorithm provides a strategy to select training samples efficiently and save computational costs by reducing the redundancy of FEM-based training samples. The adaptive algorithm calculates the residual error of the FP governing equations using automatic differentiation of the deep neural network. A probability density function expressed in terms of the residual error is used to select training samples from the Sobol sequence space. The temperature and degree of cure evolution of each training sample are obtained by a 2D FEM simulation. The adaptive method is more efficient and has a better prediction accuracy than the random sampling method. With the well-trained surrogate neural network, the FP characteristics (front speed, shape, and temperature) can be extracted quickly from the predicted temperature and degree-of-cure fields.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2021

Inference of neutrino flavor evolution through data assimilation and neural differential equations

The evolution of neutrino flavor in dense environments such as core-collapse supernovae and binary compact object mergers constitutes an important and unsolved problem. Its solution has potential implications for the dynamics and heavy-element nucleosynthesis in these environments. In this paper, we build upon recent work to explore inference-based techniques for the estimation of model parameters and neutrino flavor evolution histories. Here, we combine data assimilation, ordinary differential equation solvers, and neural networks to craft an inference approach tailored for nonlinear dynamical systems. Using this architecture, and a simple two-neutrino-beam, two-flavor model, we compare the performances of nine different optimization algorithms and expand upon previous assessments of the efficacy of inference for tackling problems in flavor evolution. We find that employing this new architecture, together with evolutionary optimization algorithms, accurately captures flavor histories in the small-scale model and allows us to quickly explore both model parameters and initial flavor content. In future work we plan to extend these inference techniques to large numbers of neutrinos.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2025

Extended Galerkin Neural Network Approximation of Singular Variational Problems with Error Control

We present extended Galerkin neural networks, a variational framework for approximating general boundary value problems (BVPs) with error control. The main contributions of this work are (1) a rigorous theory guiding the construction of new weighted least squares variational formulations suitable for use in neural network approximation of general BVPs, and (2) an “extended” feedforward network architecture which incorporates and is even capable of learning singular solution structures, thus greatly improving approximability of singular solutions. Furthermore, numerical results are presented for several problems, including steady Stokes flow around reentrant corners and in convex corners with Moffatt eddies in order to demonstrate efficacy of the method.

a posteriori error estimate↗
DOE OSTI2022

Differentiable Predictions for Large Scale Structure with SHAMNet

In simulation-based models of the galaxy-halo connection, theoretical predictions for galaxy clustering and lensing are typically made based on Monte Carlo realizations of a mock universe. In this paper, we use Subhalo Abundance Matching (SHAM) as a toy model to introduce an alternative to stochastic predictions based on mock population, demonstrating how to make simulation-based predictions for clustering and lensing that are both exact and differentiable with respect to the parameters of the model. Conventional implementations of SHAM are based on iterative algorithms such as Richardson-Lucy deconvolution; here we use the JAX library for automatic differentiation to train SHAMNet, a neural network that accurately approximates the stellar-to-halo mass relation (SMHM) defined by abundance matching. In our approach to making differentiable predictions of large-scale structure, we map parameterized PDFs onto each simulated halo, and calculate gradients of summary statistics of the galaxy distribution by using autodiff to propagate the gradients of the SMHM through the statistical estimators used to measure one- and two-point functions. Our techniques are quite general, and we conclude with an overview of how they can be applied in tandem with more complex, higher-dimensional models, creating the capability to make differentiable predictions for the multi-wavelength universe of galaxies.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2023

The role of stiffness in training and generalization of ResNets

Neural ordinary differential equations (NODEs) have recently regained popularity as large-depth limits of a large class of neural networks. In particular, residual neural networks (ResNets) are equivalent to an explicit Euler discretization of an underlying NODE, where the transition from one layer to the next is one time step of the discretization. The relationship between continuous and discrete neural networks has been of particular interest. Notably, analysis from the ordinary differential equation viewpoint can potentially lead to new insights for understanding the behavior of neural networks in general. In this work, we take inspiration from differential equations to define the concept of stiffness for a ResNet via the interpretation of a ResNet as the discretization of a NODE. Here, we then examine the effects of stiffness on the ability of a ResNet to generalize, via computational studies on example problems coming from climate and chemistry models. We find that penalizing stiffness does have a unique regularizing effect, but we see no benefit to penalizing stiffness over L 2 regularization (penalization of network parameter norms) in terms of predictive performance.

97 MATHEMATICS AND COMPUTING↗