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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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At least 91 records · Page 5
All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

A backward SDE method for uncertainty quantification in deep learning

Here, we develop a backward stochastic differential equation based probabilistic machine learning method, which formulates a class of stochastic neural networks as a stochastic optimal control problem. An efficient stochastic gradient descent algorithm is introduced with the gradient computed through a backward stochastic differential equation. Convergence analysis for stochastic gradient descent optimization and numerical experiments for applications of stochastic neural networks are carried out to validate our methodology in both theory and performance.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Do optimization methods in deep learning applications matter?

With advances in deep learning, exponential data growth and increasing model complexity, developing efficient optimization methods are attracting much research attention. Several implementations favor the use of Conjugate Gradient (CG) and Stochastic Gradient Descent (SGD) as being practical and elegant solutions to achieve quick convergence, however, these optimization processes also present many limitations in learning across deep learning applications. Recent research is exploring higher-order optimization functions as better approaches, but these present very complex computational challenges for practical use. Comparing first and higher-order optimization functions, in this paper, our experiments reveal that Levemberg-Marquardt (LM) significantly supersedes optimal convergence but suffers from very large processing time increasing the training complexity of both, classification and reinforcement learning problems. Our experiments compare off-the-shelf optimization functions(CG, SGD, LM and L-BFGS) in standard CIFAR, MNIST, CartPole and FlappyBird experiments.The paper presents arguments on which optimization functions to use and further, which functions would benefit from parallelization efforts to improve pretraining time and learning rate convergence.

Ozyildirim, Buse Melis↗
NASA NTRSDate not supplied

Robust Trajectory Optimization Techniques Using a Sweeping Gradient Method and Linear Covariance Analysis

We present robust trajectory optimization techniques using a sweeping gradient method for ordinary differential equations with events (SGM) and linear covariance analysis (LinCov). SGM is a method for computing the gradient of trajectory analyses defined by performance indices over initial value problems with events with respect to static parameters. LinCov is an analytic technique for predicting stochastic behavior of dynamical systems. By combining SGM and LinCov, it is possible use efficient, off-the-shelf, gradient-based optimizers to solve robust optimal trajectory design problems. We describe the individual methods and some details on how they can be combined. Then we apply the combined techniques to a variety of orbital trajectory design problems to demonstrate its use, including minimum fuel transfer and mid-course correction burn scheduling.

Benjamin W L Margolis↗
DOE OSTI2024

Optimization of nonlinear turbulence in stellarators

We present new stellarator equilibria that have been optimized for reduced turbulent transport using nonlinear gyrokinetic simulations within the optimization loop. The optimization routine involves coupling the pseudo-spectral GPU-native gyrokinetic code GX with the stellarator equilibrium and optimization code DESC . Since using GX allows for fast nonlinear simulations, we directly optimize for reduced nonlinear heat fluxes. To handle the noisy heat flux traces returned by these simulations, we employ the simultaneous perturbation stochastic approximation (SPSA) method that only uses two objective function evaluations for a simple estimate of the gradient. We show several examples that optimize for both reduced heat fluxes and good quasi-symmetry as a proxy for low neoclassical transport. Finally, we run full transport simulations using the T3D stellarator transport code to evaluate the changes in the macroscopic profiles.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2021

Train Like a (Var)Pro: Efficient Training of Neural Networks with Variable Projection

Deep neural networks (DNNs) have achieved state-of-the-art performance across a variety of traditional machine learning tasks, e.g., speech recognition, image classification, and segmentation. The ability of DNNs to efficiently approximate high-dimensional functions has also motivated their use in scientific applications, e.g., to solve partial differential equations and to generate surrogate models. In this paper, we consider the supervised training of DNNs, which arises in many of the above applications. We focus on the central problem of optimizing the weights of the given DNN such that it accurately approximates the relation between observed input and target data. Devising effective solvers for this optimization problem is notoriously challenging due to the large number of weights, nonconvexity, data sparsity, and nontrivial choice of hyperparameters. To solve the optimization problem more efficiently, we propose the use of variable projection (VarPro), a method originally designed for separable nonlinear least-squares problems. Our main contribution is the Gauss--Newton VarPro method (GNvpro) that extends the reach of the VarPro idea to nonquadratic objective functions, most notably cross-entropy loss functions arising in classification. These extensions make GNvpro applicable to all training problems that involve a DNN whose last layer is an affine mapping, which is common in many state-of-the-art architectures. In our four numerical experiments from surrogate modeling, segmentation, and classification, GNvpro solves the optimization problem more efficiently than commonly used stochastic gradient descent (SGD) schemes. Finally, GNvpro finds solutions that generalize well, and in all but one example better than well-tuned SGD methods, to unseen data points.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

A Kaczmarz-inspired approach to accelerate the optimization of neural network wavefunctions

Neural network wavefunctions optimized using the variational Monte Carlo method have been shown to produce highly accurate results for the electronic structure of atoms and small molecules, but the high cost of optimizing such wavefunctions prevents their application to larger systems. We propose the Subsampled Projected-Increment Natural Gradient Descent (SPRING) optimizer to reduce this bottleneck. SPRING combines ideas from the recently introduced minimum-step stochastic reconfiguration optimizer (MinSR) and the classical randomized Kaczmarz method for solving linear least-squares problems. We demonstrate that SPRING outperforms both MinSR and the popular Kronecker-Factored Approximate Curvature method (KFAC) across a number of small atoms and molecules, given that the learning rates of all methods are optimally tuned. For example, on the oxygen atom, SPRING attains chemical accuracy after forty thousand training iterations, whereas both MinSR and KFAC fail to do so even after one hundred thousand iterations.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Efficient Stochastic Inversion Using Adjoint Models and Kernel-PCA

Performing stochastic inversion on a computationally expensive forward simulation model with a high-dimensional uncertain parameter space (e.g. a spatial random field) is computationally prohibitive even with gradient information provided. Moreover, the ‘nonlinear’ mapping from parameters to observables generally gives rise to non-Gaussian posteriors even with Gaussian priors, thus hampering the use of efficient inversion algorithms designed for models with Gaussian assumptions. In this work, we propose a novel Bayesian stochastic inversion methodology, characterized by a tight coupling between a gradient-based Langevin Markov Chain Monte Carlo (LMCMC) method and a kernel principal component analysis (KPCA). This approach addresses the ‘curse-of-dimensionality’ via KPCA to identify a low-dimensional feature space within the high-dimensional and nonlinearly correlated spatial random field. Moreover, non-Gaussian full posterior probability distribution functions are estimated via an efficient LMCMC method on both the projected low-dimensional feature space and the recovered high-dimensional parameter space. We demonstrate this computational framework by integrating and adapting recent developments such as data-driven statistics-on-manifolds constructions and reduction-through-projection techniques to solve inverse problems in linear elasticity.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS2016

Hybrid Differential Dynamic Programming with Stochastic Search

Differential dynamic programming (DDP) has been demonstrated as a viable approach to low-thrust trajectory optimization, namely with the recent success of NASAs Dawn mission. The Dawn trajectory was designed with the DDP-based Static Dynamic Optimal Control algorithm used in the Mystic software. Another recently developed method, Hybrid Differential Dynamic Programming (HDDP) is a variant of the standard DDP formulation that leverages both first-order and second-order state transition matrices in addition to nonlinear programming (NLP) techniques. Areas of improvement over standard DDP include constraint handling, convergence properties, continuous dynamics, and multi-phase capability. DDP is a gradient based method and will converge to a solution nearby an initial guess. In this study, monotonic basin hopping (MBH) is employed as a stochastic search method to overcome this limitation, by augmenting the HDDP algorithm for a wider search of the solution space.

Aziz, Jonathan↗
NASA NTRS2016

Hybrid Differential Dynamic Programming with Stochastic Search

Differential dynamic programming (DDP) has been demonstrated as a viable approach to low-thrust trajectory optimization, namely with the recent success of NASA's Dawn mission. The Dawn trajectory was designed with the DDP-based Static/Dynamic Optimal Control algorithm used in the Mystic software.1 Another recently developed method, Hybrid Differential Dynamic Programming (HDDP),2, 3 is a variant of the standard DDP formulation that leverages both first-order and second-order state transition matrices in addition to nonlinear programming (NLP) techniques. Areas of improvement over standard DDP include constraint handling, convergence properties, continuous dynamics, and multi-phase capability. DDP is a gradient based method and will converge to a solution nearby an initial guess. In this study, monotonic basin hopping (MBH) is employed as a stochastic search method to overcome this limitation, by augmenting the HDDP algorithm for a wider search of the solution space.

Aziz, Jonathan↗
DOE OSTI2025

Bias-Variance Trade-Off in Physics-Informed Neural Networks with Randomized Smoothing for High-Dimensional PDEs

Physics-Informed Neural Networks (PINNs) have triggered a paradigm shift in scientific computing, leveraging mesh-free properties and robust approximation capabilities. While proving effective for low-dimensional partial differential equations (PDEs), the computational cost of PINNs remains a hurdle in high-dimensional scenarios. This is particularly pronounced when computing high-order and high-dimensional derivatives in the physics-informed loss. Randomized Smoothing PINN (RS-PINN) introduces Gaussian noise for stochastic smoothing of the original neural net model, enabling the use of Monte Carlo methods for derivative approximation, which eliminates the need for costly automatic differentiation. Despite its computational efficiency, especially in the approximation of high-dimensional derivatives, RS-PINN introduces biases in both loss and gradients, negatively impacting convergence, especially when coupled with stochastic gradient descent (SGD) algorithms. We present a comprehensive analysis of biases in RS-PINN, attributing them to the nonlinearity of the Mean Squared Error (MSE) loss as well as the intrinsic nonlinearity of the PDE itself. We propose tailored bias correction techniques, delineating their application based on the order of PDE nonlinearity. The derivation of an unbiased RS-PINN allows for a detailed examination of its advantages and disadvantages compared to the biased version. Specifically, the biased version has a lower variance and runs faster than the unbiased version, but it is less accurate due to the bias. To optimize the bias-variance trade-off, we combine the two approaches in a hybrid method that balances the rapid convergence of the biased version with the high accuracy of the unbiased version. In addition to methodological contributions, we present an enhanced implementation of RS-PINN. Extensive experiments on diverse high-dimensional PDEs, including Fokker-Planck, Hamilton-Jacobi-Bellman (HJB), viscous Burgers’, Allen-Cahn, and Sine-Gordon equations, illustrate the bias-variance trade-off and highlight the effectiveness of the hybrid RS-PINN. Empirical guidelines are provided for selecting biased, unbiased, or hybrid versions, depending on the dimensionality and nonlinearity of the specific PDE problem.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Random coordinate descent: A simple alternative for optimizing parameterized quantum circuits

Variational quantum algorithms rely on the optimization of parameterized quantum circuits in noisy settings. The commonly used back-propagation procedure in classical machine learning is not directly applicable in this setting due to the collapse of quantum states after measurements. Thus, gradient estimations constitute a significant overhead in a gradient-based optimization of such quantum circuits. This paper introduces a random coordinate descent algorithm as a practical and easy-to-implement alternative to the full gradient descent algorithm. This algorithm only requires one partial derivative at each iteration. Motivated by the behavior of measurement noise in the practical optimization of parameterized quantum circuits, this paper presents an optimization problem setting that is amenable to analysis. Under this setting, the random coordinate descent algorithm exhibits the same level of stochastic stability as the full gradient approach, making it as resilient to noise. The complexity of the random coordinate descent method is generally no worse than that of the gradient descent and can be much better for various quantum optimization problems with anisotropic Lipschitz constants. Theoretical analysis and extensive numerical experiments validate our findings. Published by the American Physical Society 2024

Ding, Zhiyan (ORCID:000000018863403X)↗
DOE OSTI2020

Locally adaptive activation functions with slope recovery for deep and physics-informed neural networks

Here we propose two approaches of locally adaptive activation functions namely, layer-wise and neuron-wise locally adaptive activation functions, which improve the performance of deep and physics-informed neural networks. The local adaptation of activation function is achieved by introducing a scalable parameter in each layer (layer-wise) and for every neuron (neuron-wise) separately, and then optimizing it using a variant of stochastic gradient descent algorithm. In order to further increase the training speed, an activation slope-based slope recovery term is added in the loss function, which further accelerates convergence, thereby reducing the training cost. On the theoretical side, we prove that in the proposed method, the gradient descent algorithms are not attracted to sub-optimal critical points or local minima under practical conditions on the initialization and learning rate, and that the gradient dynamics of the proposed method is not achievable by base methods with any (adaptive) learning rates. We further show that the adaptive activation methods accelerate the convergence by implicitly multiplying conditioning matrices to the gradient of the base method without any explicit computation of the conditioning matrix and the matrix–vector product. The different adaptive activation functions are shown to induce different implicit conditioning matrices. Furthermore, the proposed methods with the slope recovery are shown to accelerate the training process.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Global stochastic optimization of stellarator coil configurations

In the construction of a stellarator, the manufacturing and assembling of the coil system is a dominant cost. These coils need to satisfy strict engineering tolerances, and if those are not met the project could be cancelled as in the case of the National Compact Stellarator Experiment (NCSX) project. Therefore, our goal is to find coil configurations that increase construction tolerances without compromising the performance of the magnetic field. In this paper, we develop a gradient-based stochastic optimization model which seeks robust stellarator coil configurations in high dimensions. In particular, we design a two-step method: first, we perform an approximate global search by a sample efficient trust-region Bayesian optimization; second, we refine the minima found in step one with a stochastic local optimizer. To this end, we introduce two stochastic local optimizers: BFGS applied to the sample average approximation; and Adam, equipped with a control variate for variance reduction. Numerical simulations performed on a W7-X-like coil configuration demonstrate that our global optimization approach finds a variety of promising local solutions at less than 0.1% of the cost of previous work, which considered solely local stochastic optimization.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
NASA NTRSDate not supplied

Gradient-Based Optimization of the Common Research Model Wing Subject to CFD-Based Gust and Flutter Constraints

The linearized frequency-domain method was recently implemented in the stabilized finite element solver in NASA’s FUN3D code. Previous work by the authors used this method for enforcing flutter constraints during gradient-based optimizations. More recently, the solver was expanded to account for continuous (also known as stochastic) gust responses. This paper expands on recent Common Research Model wing optimization work, which demonstrated gradient-based optimization with flutter and stochastic gust constraints, among others. While that work utilized FUN3D for static aeroelastic solutions but relied on doublet lattice aerodynamics for gust and flutter responses, the present work replaces these unsteady aerodynamic analyses with those of FUN3D’s linearized frequency-domain solver. With analytic derivatives available, gradient-based optimization is performed through the use of the OpenMDAO/MPhys libraries with over 700 shape, structural, and aerodynamic design variables and over 10 nonlinear constraints. Comparisons of analysis results and optimized designs are made between doublet lattice and linearized frequency-domain solutions.

aeroelasticity↗
NASA NTRS1993

Statistics of Experiments on Cluster Formation and Transport in a Gravitational Field

Metastable state relaxation in a gravitational field is investigated in the case of non-critical binary solutions. A relaxation description is presented in terms of the time-dependent Ginzburg-Landau formalism for a non-conserved order parameter. A new ansatz for solution of the corresponding partial nonlinear stochastic differential equation is discussed. It is proved that, for the supersaturated solution under consideration, the metastable state relaxation in a gravitational field leads to formation of solute concentration gradients due to the sedimentation of subcritical solute clusters. The pure discussion of the possible methods to compare theoretical results and experimental data related to solute sedimentation in a gravitational field is presented. It is shown that in order to describe these experiments it is necessary to deal both with the value of the solute concentration gradient and with its formation rate. The stochastic nature of the sedimentation process is shown.

Izmailov, Alexander F.↗
DOE OSTI2023

Stochastic projective splitting

Here, we present a new, stochastic variant of the projective splitting (PS) family of algorithms for inclusion problems involving the sum of any finite number of maximal monotone operators. This new variant uses a stochastic oracle to evaluate one of the operators, which is assumed to be Lipschitz continuous, and (deterministic) resolvents to process the remaining operators. Our proposal is the first version of PS with such stochastic capabilities. We envision the primary application being machine learning (ML) problems, with the method’s stochastic features facilitating “mini-batch” sampling of datasets. Since it uses a monotone operator formulation, the method can handle not only Lipschitz-smooth loss minimization, but also min–max and noncooperative game formulations, with better convergence properties than the gradient descent-ascent methods commonly applied in such settings. The proposed method can handle any number of constraints and nonsmooth regularizers via projection and proximal operators. We prove almost-sure convergence of the iterates to a solution and a convergence rate result for the expected residual, and close with numerical experiments on a distributionally robust sparse logistic regression problem.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Accelerating Hamiltonian Monte Carlo for Bayesian inference in neural networks and neural operators

Hamiltonian Monte Carlo (HMC) is a powerful and accurate method to sample from the posterior distribution in Bayesian inference. However, HMC techniques are computationally demanding for Bayesian neural networks due to the high dimensionality of the network’s parameter space and the non-convexity of their posterior distributions. Therefore, various approximation techniques, such as variational inference (VI) or stochastic gradient MCMC, are often employed to infer the posterior distribution of the network parameters. Such approximations introduce inaccuracies in the inferred distributions, resulting in unreliable uncertainty estimates. In this work, we propose a hybrid approach that combines inexpensive VI and accurate HMC methods to efficiently and accurately quantify uncertainties in neural networks and neural operators. The proposed approach leverages an initial VI training on the full network. We examine the influence of individual parameters on the prediction uncertainty, which shows that a large proportion of the parameters do not contribute substantially to uncertainty in the network predictions. This information is then used to significantly reduce the dimension of the parameter space, and HMC is performed only for the subset of network parameters that strongly influence prediction uncertainties. This yields a framework for accelerating the full batch HMC for posterior inference in neural networks. We demonstrate the efficiency and accuracy of the proposed framework on deep neural networks and operator networks, showing that inference can be performed for large networks with tens to hundreds of thousands of parameters. Finally, we show that this method can effectively learn surrogates for complex physical systems by modeling the operator that maps from upstream conditions to wall-pressure data on a cone in hypersonic flow.

Bayesian inference↗
DOE OSTI2019

An Efficient, Globally Convergent Method for Optimization Under Uncertainty Using Adaptive Model Reduction and Sparse Grids

We present a new approach to efficiently solve optimization problems constrained by partial differential equations (PDEs) with uncertain coefficients. The method leverages two sources of inexactness that trade accuracy for speed: (1) stochastic collocation based on dimension-adaptive sparse grids (SGs), which approximates the stochastic objective function with a limited number of quadrature nodes, and (2) projection-based reduced-order models (ROMs), which generate efficient approximations to PDE solutions. These two sources of inexactness lead to inexact objective function and gradient evaluations, which are managed by a trust-region method that guarantees global convergence by adaptively refining the SG and ROM until a proposed error indicator drops below a tolerance specified by trust-region convergence theory. A key feature of the proposed method is that the error indicator---which accounts for errors incurred by both the SG and ROM---must be only an asymptotic error bound, i.e., a bound that holds up to an arbitrary constant that need not be computed. This allows the approach to be applicable to a wide range of problems, including those where sharp, computable error bounds are not available; this distinguishes the proposed method from previous works. Numerical experiments performed on a model problem from optimal flow control under uncertainty verify global convergence of the method and demonstrate the method's ability to outperform previously proposed alternatives.

97 MATHEMATICS AND COMPUTING↗