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DOE OSTI2023

Structural characterization of an intrinsically disordered protein complex using integrated small-angle neutron scattering and computing

Characterizing structural ensembles of intrinsically disordered proteins (IDPs) and intrinsically disordered regions (IDRs) of proteins is essential for studying structure–function relationships. Due to the different neutron scattering lengths of hydrogen and deuterium, selective labeling and contrast matching in small-angle neutron scattering (SANS) becomes an effective tool to study dynamic structures of disordered systems. However, experimental timescales typically capture measurements averaged over multiple conformations, leaving complex SANS data for disentanglement. We hereby demonstrate an integrated method to elucidate the structural ensemble of a complex formed by two IDRs. We use data from both full contrast and contrast matching with residue-specific deuterium labeling SANS experiments, microsecond all-atom molecular dynamics (MD) simulations with four molecular mechanics force fields, and an autoencoder-based deep learning (DL) algorithm. From our combined approach, we show that selective deuteration provides additional information that helps characterize structural ensembles. We find that among the four force fields, a99SB-disp and CHARMM36m show the strongest agreement with SANS and NMR experiments. In addition, our DL algorithm not only complements conventional structural analysis methods but also successfully differentiates NMR and MD structures which are indistinguishable on the free energy surface. Finally, we present an ensemble that describes experimental SANS and NMR data better than MD ensembles generated by one single force field and reveal three clusters of distinct conformations. Our results demonstrate a new integrated approach for characterizing structural ensembles of IDPs.

59 BASIC BIOLOGICAL SCIENCES↗

DOE OSTI2022

Approximating the linear response of physical chaos

Abstract Parametric derivatives of statistics are highly desired quantities in prediction, design optimization and uncertainty quantification. In the presence of chaos, the rigorous computation of these quantities is certainly possible, but mathematically complicated and computationally expensive. Based on Ruelle’s formalism, this paper shows that the sophisticated linear response algorithm can be dramatically simplified in higher-dimensional systems featuring statistical homogeneity in the physical space. We argue that the contribution of the SRB (Sinai–Ruelle–Bowen) measure gradient, which is an integral yet the most cumbersome part of the full algorithm, is negligible if the objective function is appropriately aligned with unstable manifolds. This abstract condition could potentially be satisfied by a vast family of real-world chaotic systems, regardless of the physical meaning and mathematical form of the objective function and perturbed parameter. We demonstrate several numerical examples that support these conclusions and that present the use and performance of a simplified linear response algorithm. In the numerical experiments, we consider physical models described by differential equations, including Lorenz 96 and Kuramoto–Sivashinsky.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2021

Linear embedding of nonlinear dynamical systems and prospects for efficient quantum algorithms

The simulation of large nonlinear dynamical systems, including systems generated by discretization of hyperbolic partial differential equations, can be computationally demanding. Such systems are important in both fluid and kinetic computational plasma physics. This motivates exploring whether a future error-corrected quantum computer could perform these simulations more efficiently than any classical computer. In this work, we describe a method for mapping any finite nonlinear dynamical system to an infinite linear dynamical system (embedding) and detail three specific cases of this method that correspond to previously studied mappings. Then we explore an approach for approximating the resulting infinite linear system with finite linear systems (truncation). Using a number of qubits only logarithmic in the number of variables of the nonlinear system, a quantum computer could simulate truncated systems to approximate output quantities if the nonlinearity is sufficiently weak. Other aspects of the computational efficiency of the three detailed embedding strategies are also discussed.

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 OSTI2025

Engineering Privacy at the Edge: A Practical Guide to Differential Privacy in System Architectures

The rapid expansion of distributed and edge computing platforms—spanning autonomous vehicles, IoT sensors, and healthcare monitors—has heightened concerns about data privacy. Differential Privacy (DP) offers a rigorous mathematical framework to protect sensitive information while retaining analytical utility. This tutorial introduces the foundations of DP for both numerical and categorical datasets and extends the discussion to correlation-aware techniques tailored for structured and high-dimensional data. Hands-on demonstrations will begin with the PETINA (Privacy prEservaTIoN Algorithms) package for numerical data and continue with MIC-DP (Maximum Information Correlated Differential Privacy) for tabular data. Designed for researchers and practitioners in secure systems, embedded architectures, and AI accelerators, the tutorial emphasizes practical and scalable methods for integrating DP into real-world system designs.

Kotevska, Olivera [ORNL] (ORCID:0000000316772243)↗

DOE OSTI2022

Differential methods for assessing sensitivity in biological models

Differential sensitivity analysis is indispensable in fitting parameters, understanding uncertainty, and forecasting the results of both thought and lab experiments. Although there are many methods currently available for performing differential sensitivity analysis of biological models, it can be difficult to determine which method is best suited for a particular model. In this paper, we explain a variety of differential sensitivity methods and assess their value in some typical biological models. First, we explain the mathematical basis for three numerical methods: adjoint sensitivity analysis, complex perturbation sensitivity analysis, and forward mode sensitivity analysis. We then carry out four instructive case studies. (a) The CARRGO model for tumor-immune interaction highlights the additional information that differential sensitivity analysis provides beyond traditional naive sensitivity methods, (b) the deterministic SIR model demonstrates the value of using second-order sensitivity in refining model predictions, (c) the stochastic SIR model shows how differential sensitivity can be attacked in stochastic modeling, and (d) a discrete birth-death-migration model illustrates how the complex perturbation method of differential sensitivity can be generalized to a broader range of biological models. Finally, we compare the speed, accuracy, and ease of use of these methods. We find that forward mode automatic differentiation has the quickest computational time, while the complex perturbation method is the simplest to implement and the most generalizable.

59 BASIC BIOLOGICAL SCIENCES↗

DOE OSTI2020

Adaptive Activation Functions Accelerate Convergence in Deep and Physics-informed Neural Networks

We employ adaptive activation functions for regression in deep and physics-informed neural networks (PINNs) to approximate smooth and discontinuous functions as well as solutions of linear and nonlinear partial differential equations. In particular, we solve the nonlinear Klein-Gordon equation, which has smooth solutions, the nonlinear Burgers equation, which can admit high gradient solutions, and the Helmholtz equation. We introduce a scalable hyper-parameter in the activation function, which can be optimized to achieve best performance of the network as it changes dynamically the topology of the loss function involved in the optimization process. The adaptive activation function has better learning capabilities than the traditional one (fixed activation) as it improves greatly the convergence rate, especially at early training, as well as the solution accuracy. To better understand the learning process, we plot the neural network solution in the frequency domain to examine how the network captures successively different frequency bands present in the solution. We consider both forward problems, where the approximate solutions are obtained, as well as inverse problems, where parameters involved in the governing equation are identified. Our simulation results show that the proposed method is a very simple and effective approach to increase the efficiency, robustness and accuracy of the neural network approximation of nonlinear functions as well as solutions of partial differential equations, especially for forward problems. We theoretically prove that in the proposed method, gradient descent algorithms are not attracted to suboptimal critical points or local minima.

machine leaning, Bad minima, Inverse problems, Phy↗

DOE OSTI2023

A greedy Galerkin method to efficiently select sensors for linear dynamical systems

A key challenge in inverse problems is the selection of sensors to gather the most effective data. In this paper, we consider the problem of inferring the initial condition to a linear dynamical system and develop an efficient control-theoretical approach for greedily selecting sensors. Our method employs a Galerkin projection to reduce the size of the inverse problem, resulting in a computationally efficient algorithm for sensor selection. As a byproduct of our algorithm, we obtain a preconditioner for the inverse problem that enables the rapid recovery of the initial condition. Here, we analyze the theoretical performance of our greedy sensor selection algorithm as well as the performance of the associated preconditioner. Finally, we verify our theoretical results on various inverse problems involving partial differential equations.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

Geometric Interpretation of the Cluster Location Problem Part I: Theory

We present a new framing of the seismic location problem using principles drawn from differential geometry. Our interpretation relies upon the common assumption that travel times observed across a network are continuous, differentiable functions of source location. In consequence, travel‐time functions constitute a differentiable map between the source region and a Riemannian manifold. The manifold is said to be the image of the source region embedded in a generally high‐dimension travel‐time vector space. A cluster of events in the source region has an image of discrete points on the manifold, that, except in the simplest cases, cannot be viewed directly. However, it is possible to project the image of a cluster into a tangent space of the manifold for direct visualization. The projection operator can be computed directly from the data without a velocity model, but produces a distorted rendering of the cluster geometry. With a model we can predict the distortions and correct them to estimate cluster geometry. We develop these points with the simplest possible example, one for which direct visualization of the manifold is possible, using the example as an introduction to the relevant concepts from differential geometry in a familiar setting. The tangent space, a local linearization of the manifold, plays a key role. We develop a metric to estimate the limits of linearization, that is, to determine when the curvature of the manifold invalidates the linear assumption. We also examine the interplay of model error, inadequate network geometry, and pick error. We then generalize our results from the simple case to the general case of 3D source regions observed by general networks. Although we do suggest a new “project and correct” method for location, we do not develop it into a practical algorithm. In conclusion, our intention rather is to highlight new analytical methods grounded in differential geometry.

East Pacific Ocean Islands↗

DOE OSTI2021

Tusas: A fully implicit parallel approach for coupled phase-field equations

In this study, we develop a fully-coupled, fully-implicit approach for phase-field modeling of solidification in metals and alloys. Predictive simulation of solidification in pure metals and metal alloys remains a significant challenge in the field of materials science, as microstructure formation during the solidification process plays a critical role in the properties and performance of the solid material. Our simulation approach consists of a finite element spatial discretization of the fully-coupled nonlinear system of partial differential equations at the microscale, which is treated implicitly in time with a preconditioned Jacobian-free Newton-Krylov method. The approach is algorithmically scalable as well as efficient due to an effective preconditioning strategy based on algebraic multigrid and block factorization. We implement this approach in the open-source Tusas framework, which is a general, flexible tool developed in C++ for solving coupled systems of nonlinear partial differential equations. The performance of our approach is analyzed in terms of algorithmic scalability and efficiency, while the computational performance of Tusas is presented in terms of parallel scalability and efficiency on emerging heterogeneous architectures. We demonstrate that modern algorithms, discretizations, and computational science, and heterogeneous hardware provide a robust route for predictive phase-field simulation of microstructure evolution during additive manufacturing.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

nPINNs: Nonlocal physics-informed neural networks for a parametrized nonlocal universal Laplacian operator. Algorithms and applications

Physics-informed neural networks (PINNs) are effective in solving inverse problems based on differential and integro-differential equations with sparse, noisy, unstructured, and multifidelity data. PINNs incorporate all available information, including governing equations (reflecting physical laws), initial-boundary conditions, and observations of quantities of interest, into a loss function to be minimized, thus recasting the original problem into an optimization problem. In this paper, we extend PINNs to parameter and function inference for integral equations such as nonlocal Poisson and nonlocal turbulence models, and we refer to them as nonlocal PINNs (nPINNs). The contribution of the paper is three-fold. First, we propose a unified nonlocal Laplace operator, which converges to the classical Laplacian as one of the operator parameters, the nonlocal interaction radius $\delta$ goes to zero, and to the fractional Laplacian as $\delta$ goes to infinity. This universal operator forms a super-set of classical Laplacian and fractional Laplacian operators and, thus, has the potential to fit a broad spectrum of data sets. We also provide theoretical convergence rates with respect to $\delta$ and verify them via numerical experiments. Second, we use nPINNs to estimate the two parameters, $\delta$ and $\alpha$, characterizing the kernel of the unified operator. The strong non-convexity of the loss function yielding multiple (good) local minima reveals the occurrence of the operator mimicking phenomenon, that is, different pairs of estimated parameters could produce multiple solutions of comparable accuracy. Third, we propose another nonlocal operator with spatially variable order $\alpha(y)$, which is more suitable for modeling turbulent Couette flow. Our results show that nPINNs can jointly infer this function as well as $\delta$. More importantly, these parameters exhibit a universal behavior with respect to the Reynolds number, a finding that contributes to our understanding of nonlocal interactions in wall-bounded turbulence.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

nPINNs: nonlocal Physics-Informed Neural Networks for a parametrized nonlocal universal Laplacian operator. Algorithms and Applications

Physics-informed neural networks (PINNs) are effective in solving inverse problems based on differential and integro-differential equations with sparse, noisy, unstructured, and multifidelity data. PINNs incorporate all available information, including governing equations (reflecting physical laws), initial-boundary conditions, and observations of quantities of interest, into a loss function to be minimized, thus recasting the original problem into an optimization problem. In this paper, we extend PINNs to parameter and function inference for integral equations such as nonlocal Poisson and nonlocal turbulence models, and we refer to them as nonlocal PINNs (nPINNs). The contribution of the paper is three-fold. First, we propose a unified nonlocal Laplace operator, which converges to the classical Laplacian as one of the operator parameters, the nonlocal interaction radius δ goes to zero, and to the fractional Laplacian as δ goes to infinity. This universal operator forms a super-set of classical Laplacian and fractional Laplacian operators and, thus, has the potential to fit a broad spectrum of data sets. We provide theoretical convergence rates with respect to δ and verify them via numerical experiments. Second, we use nPINNs to estimate the two parameters, δ and α, characterizing the kernel of the unified operator. The strong non-convexity of the loss function yielding multiple (good) local minima reveals the occurrence of the operator mimicking phenomenon, that is, different pairs of estimated parameters could produce multiple solutions of comparable accuracy. Third, we propose another nonlocal operator with spatially variable order α(γ), which is more suitable for modeling turbulent Couette flow. Our results show that nPINNs can jointly infer this function as well as δ. More importantly, these parameters exhibit a universal behavior with respect to the Reynolds number, a finding that contributes to our understanding of nonlocal interactions in wall-bounded turbulence.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2023

Domain Aware Deep-learning Algorithms Integrated with Scientific-computing Technologies (DADAIST)

This technical report summarized the contribution of the DADAIST project funded by the Data Model Convergence Initiative via the Laboratory Directed Research and Development (LDRD) investments at Pacific Northwest National Laboratory (PNNL). Specifically, we report the development of the NeuroMANCER (Neural Modules with Adaptive Nonlinear Constraints and Efficient Regularizations), a new open-source Scientific Machine Learning library for formulating and solving parametric constrained optimization problems, physics-informed system identification, and parametric optimal control problems. NeuroMANCER is using differentiable programming to combine modern data-driven models and optimization modeling language into a coherent algorithmic and software framework. NeuroMANCER is a Pytorch-based framework and adopts much of its philosophy focused on research and development, rapid prototyping, and streamlined deployment. Strong emphasis is given to extensibility, interoperability with the PyTorch ecosystem, and quick adaptability to custom domain problems. Neuromancer repository contains a comprehensive library of differentiable modules, including custom activation functions, matrix factorizations, deep learning architectures, neural differential equations, differential equation solvers, implicit layers such as iterative solvers, high-level API for symbolic expressions, API for modeling and control of dynamical systems, and extensive set of tutorial code examples in the form of python scripts and jupyter notebooks.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

Measurement of inclusive charged-particle jet production in pp and p-Pb collisions at $ \sqrt{s_{\textrm{NN}}} $ = 5.02 TeV

Measurements of inclusive charged-particle jet production in pp and p-Pb collisions at center-of-mass energy per nucleon-nucleon collision $\sqrt{^sNN}$ = 5.02 TeV and the corresponding nuclear modification factor $R^{ch jet}_{pPb}$ are presented, using data collected with the ALICE detector at the LHC. Jets are reconstructed in the central rapidity region |η jet | < 0.5 from charged particles using the anti-k T algorithm with resolution parameters R = 0.2, 0.3, and 0.4. The p T -differential inclusive production cross section of charged-particle jets, as well as the corresponding cross section ratios, are reported for pp and p-Pb collisions in the transverse momentum range 10 < $p^{ch}_{T,jet}$ < 140 GeV/c and 10 < $p^{ch}_{T,jet}$ < 160 GeV/c, respectively, together with the nuclear modification factor $R^{ch jet}_{pPb}$ in the range 10 < $p^{ch}_{T,jet}$ < 140 GeV/c. The analysis extends the p T range of the previously-reported charged-particle jet measurements by the ALICE Collaboration. The nuclear modification factor is found to be consistent with one and independent of the jet resolution parameter with the improved precision of this study, indicating that the possible influence of cold nuclear matter effects on the production cross section of charged-particle jets in p-Pb collisions at $\sqrt{^sNN}$ = 5.02 TeV is smaller than the current precision. The obtained results are in agreement with other minimum bias jet measurements available for RHIC and LHC energies, and are well reproduced by the NLO perturbative QCD Powheg calculations with parton shower provided by PYTHIA8 as well as by JETSCAPE simulations.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2021

Physics-informed machine learning

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. Furthermore, 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.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

Applying Quantum Computing to Simulate Power System Dynamics

Power system dynamics are generally modeled by high dimensional nonlinear differential-algebraic equations due to a large number of generators, loads, and transmission lines. Thus, its computational complexity grows exponentially with the system size. This paper demonstrates the potential use of quantum computing algorithms to model the power system dynamics. Leveraging a symbolic programming framework, we equivalently convert the power system dynamics’ differential algebraic equations (DAEs) into ordinary differential equations (ODEs), where the data of the state vector can be encoded into quantum computers via amplitude encoding. The system's nonlinearity is captured by Taylor polynomial expansion, the quantum state tensor, and Hamiltonian simulation, whereas state variables can be updated by a quantum linear equation solver. Our results show that quantum computing can simulate the dynamics of the power system with high accuracy, whereas its complexity is polynomial in the logarithm of the system dimension. Our work also illustrates the use of scientific machine learning tools for implementing scientific computing concepts, e.g., Taylor expansion, DAEs/ODEs transform, and quantum computing solver, in the field of power engineering.

Tran, Huynh↗

DOE OSTI2024

Metric Learning to Accelerate Convergence of Operator Splitting Methods

Recent developments in machine learning have led to promising advances in accelerating the solution of constrained optimization problems. Increasing demand for real-time decision-making capabilities in applications such as artificial intelligence and optimal control has led to a variety of proposed strategies for learning to produce fast solutions to optimization problems. For example, recent works have shown that it is possible to accelerate the convergence of optimization algorithms by learning to select their parameters, such as gradient descent stepsizes. This work proposes a new approach, in which the underlying metric spaces of proximal operator splitting algorithms are learned to maximize convergence rate. While prior works in optimization theory have derived optimal metrics in simple cases, no such result exists for many practical problem forms including general Quadratic Programming (QP). This paper shows how differentiable optimization can enable the end-to-end learning of proximal metrics, enhancing the convergence of proximal algorithms for QP problems beyond what is possible based on known theory. Additionally, the results illustrate a strong connection between the learned proximal metrics and active constraints at the optima, leading to an interpretation in which the predicted proximal metrics can be viewed as a form of active set prediction.

King, Ethan [BATTELLE (PACIFIC NW LAB)]↗

DOE OSTI2022

A phase transition for finding needles in nonlinear haystacks with LASSO artificial neural networks

To fit sparse linear associations, a LASSO sparsity inducing penalty with a single hyperparameter provably allows to recover the important features (needles) with high probability in certain regimes even if the sample size is smaller than the dimension of the input vector (haystack). More recently learners known as artificial neural networks (ANN) have shown great successes in many machine learning tasks, in particular fitting nonlinear associations. Small learning rate, stochastic gradient descent algorithm and large training set help to cope with the explosion in the number of parameters present in deep neural networks. Yet few ANN learners have been developed and studied to find needles in nonlinear haystacks. Driven by a single hyperparameter, our ANN learner, like for sparse linear associations, exhibits a phase transition in the probability of retrieving the needles, which we do not observe with other ANN learners. To select our penalty parameter, we generalize the universal threshold of Donoho and Johnstone (Biometrika 81(3):425–455, 1994) which is a better rule than the conservative (too many false detections) and expensive cross-validation. In the spirit of simulated annealing, we propose a warm-start sparsity inducing algorithm to solve the high-dimensional, non-convex and non-differentiable optimization problem. We perform simulated and real data Monte Carlo experiments to quantify the effectiveness of our approach.

97 MATHEMATICS AND COMPUTING↗