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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 OSTI2025

Constrained or unconstrained? Neural-network-based equation discovery from data

Throughout many fields, practitioners often rely on differential equations to model systems. Yet, for many applications, the theoretical derivation of such equations and/or the accurate resolution of their solutions may be intractable. Instead, recently developed methods, including those based on parameter estimation, operator subset selection, and neural networks, allow for the data-driven discovery of both ordinary and partial differential equations (PDEs), on a spectrum of interpretability. The success of these strategies is often contingent upon the correct identification of representative equations from noisy observations of state variables and, as importantly and intertwined with that, the mathematical strategies utilized to enforce those equations. Specifically, the latter has been commonly addressed via unconstrained optimization strategies. Representing the PDE as a neural network, we propose to discover the PDE (or the associated operator) by solving a constrained optimization problem and using an intermediate state representation similar to a physics-informed neural network (PINN). The objective function of this constrained optimization problem promotes matching the data, while the constraints require that the discovered PDE is satisfied at a number of spatial collocation points. We present a penalty method and a widely used trust-region barrier method to solve this constrained optimization problem, and we compare these methods on numerical examples. Our results on several example problems demonstrate that the latter constrained method outperforms the penalty method, particularly for higher noise levels or fewer collocation points. This work motivates further exploration into using sophisticated constrained optimization methods in scientific machine learning, as opposed to their commonly used, penalty-method or unconstrained counterparts. For both of these methods, we solve these discovered neural network PDEs with classical methods, such as finite difference methods, as opposed to PINNs-type methods relying on automatic differentiation. Here, we briefly highlight how simultaneously fitting the data while discovering the PDE improves the robustness to noise and other small, yet crucial, implementation details.

Data-driven discovery↗
DOE OSTI2026

SODAs: sparse optimization for the discovery of differential and algebraic equations

Differential-algebraic equations (DAEs) integrate ordinary differential equations (ODEs) with algebraic constraints, providing a fundamental framework for developing models of dynamical systems characterized by time-scale separation, conservation laws and physical constraints. While sparse optimization has revolutionized model development by allowing data-driven discovery of parsimonious models from a library of possible equations, existing approaches for dynamical systems assume DAEs can be reduced to ODEs by eliminating variables before model discovery. This assumption limits the applicability of such methods for DAE systems with unknown constraints and time scales. We introduce sparse optimization for differential-algebraic systems (SODAs), a data-driven method for the identification of DAEs in their explicit form. By discovering the algebraic and dynamic components sequentially without prior identification of the algebraic variables, this approach leads to a sequence of convex optimization problems. It has the advantage of discovering interpretable models that preserve the structure of the underlying physical system. To this end, SODAs improves since SODAs is singular numerical stability when handling high correlations between library terms, caused by near-perfect algebraic relationships, by iteratively refining the conditioning of the candidate library. We demonstrate the performance of our method on biological, mechanical and electrical systems, showcasing its robustness to noise in both simulated time series and real-time experimental data.

DAE↗
DOE OSTI2025

Guiding center equations for the magnetic dipole

Since the discovery of Van Allen radiation belts in the 1960s, observations of energetic ions trapped in the Earth's dipole magnetic field have illustrated the remarkable confinement properties of this configuration. As such, it has been used for confining a hot plasma for nuclear fusion studies, starting from the pioneering work of Bo Lehnert and Akira Hasegawa, in the Levitated Dipole Experiment (LDX) at MIT until 2011 and in the RT-1 experiment at the University of Tokyo. More recently, the dipole has been subject to a renewed interest for fusion studies by a couple of startups and for smaller applications as a cold plasma source. While the equilibrium and magneto-hydrodynamic stability of the dipole have been investigated quite in detail, neoclassical properties of the dipole are comparatively much less known: the dipole is more known in geophysics than in fusion science. For this reason, in this paper, we propose a set of Hamiltonian, guiding-center equations to describe the motion of electrons and ions in a magnetic dipole configuration. We also developed a code, and we show the main features of particle motion, benchmarking our results with the analytical solutions for the bounce and precession motion, which are well documented in the literature. We also draw some general conclusions for the neoclassical transport in usual toroidal confinement schemes, such as the tokamak and the stellarator, pointing out the unique advantages of the dipole in confining energetic particles.

Hamiltonian mechanics↗
DOE OSTI2025

Solving sparse finite element problems on neuromorphic hardware

The finite element method (FEM) is one of the most important and ubiquitous numerical methods for solving partial differential equations (PDEs) on computers for scientific and engineering discovery. Applying the FEM to larger and more detailed scientific models has driven advances in high-performance computing for decades. Here we demonstrate that scalable spiking neuromorphic hardware can directly implement the FEM by constructing a spiking neural network that solves the large, sparse, linear systems of equations at the core of the FEM. We show that for the Poisson equation, a fundamental PDE in science and engineering, our neural circuit achieves meaningful levels of numerical accuracy and close to ideal scaling on modern, inherently parallel and energy-efficient neuromorphic hardware, specifically Intel’s Loihi 2 neuromorphic platform. We illustrate extensions to irregular mesh geometries in both two and three dimensions as well as other PDEs such as linear elasticity. Our spiking neural network is constructed from a recurrent network model of the brain’s motor cortex and, in contrast to black-box deep artificial neural network-based methods for PDEs, directly translates the well-understood and trusted mathematics of the FEM to a natively spiking neuromorphic algorithm.

Applied mathematics↗
DOE OSTI2025

Weak Form Scientific Machine Learning: Test Function Construction for System Identification

Weak form Scientific Machine Learning (WSciML) is a recently developed framework for data-driven modeling and scientific discovery. It leverages the weak form of equation error residuals to provide enhanced noise robustness in system identification via convolving model equations with test functions, reformulating the problem to avoid direct differentiation of data. The performance, however, relies on wisely choosing a set of compactly supported test functions. In this work, we mathematically motivate a novel data-driven method for constructing Single-scale-Local reference functions for creating the set of test functions. Our approach numerically approximates the integration error introduced by the quadrature and identifies the support size for which the error is minimal, without requiring access to the model parameter values. Through numerical experiments across various models, noise levels, and temporal resolutions, we demonstrate that the selected supports consistently align with regions of minimal parameter estimation error. We also compare the proposed method against the strategy for constructing Multi-scale-Global (and orthogonal) test functions introduced in our prior work, demonstrating the improved computational efficiency.

FOS: Computer and information sciences↗
DOE OSTI2024

Asymptotic consistency of the WSINDy algorithm in the limit of continuum data

In this work we study the asymptotic consistency of the weak-form sparse identification of nonlinear dynamics algorithm (WSINDy) in the identification of differential equations from noisy samples of solutions. We prove that the WSINDy estimator is unconditionally asymptotically consistent for a wide class of models that includes the Navier–Stokes, Kuramoto–Sivashinsky and Sine–Gordon equations. We thus provide a mathematically rigorous explanation for the observed robustness to noise of weak-form equation learning. Conversely, we also show that, in general, the WSINDy estimator is only conditionally asymptotically consistent, yielding discovery of spurious terms with probability one if the noise level exceeds a critical threshold σ c . We provide explicit bounds on σ c in the case of Gaussian white noise and we explicitly characterize the spurious terms that arise in the case of trigonometric and/or polynomial libraries. Furthermore, we show that, if the data is suitably denoised (a simple moving average filter is sufficient), then asymptotic consistency is recovered for models with locally-Lipschitz, polynomial-growth nonlinearities. Our results reveal important aspects of weak-form equation learning, which may be used to improve future algorithms. We demonstrate our findings numerically using the Lorenz system, the cubic oscillator, a viscous Burgers-growth model and a Kuramoto–Sivashinsky-type high-order PDE.

asymptotic consistency↗
DOE OSTI2025

A neural master equation framework for multiscale modeling of molecular processes: application to atomic-scale plasma processes

Plasma-surface interactions (PSI) play a crucial role in microelectronics fabrication; however, their multiscale nature and array of complex, often unknown interactions make computational modeling of PSIs extremely difficult. To this end, we propose a general neural master equation (NME) framework that uses master equations to describe the dynamics of a molecular process, wherein neural networks learned from atomistic simulations represent unknown transitions between different system states. By leveraging the physics-based structure of master equations and data-driven state transitions, the NME framework promotes generalizability and physics interpretability, and can bridge disparate length and time scales. The framework is demonstrated for multiscale modeling of Si atomic layer etching and reactive ion etching, where the learned NME-based surface kinetic models exhibit good predictive and extrapolative capabilities for predicting experimentally relevant observables as a function of process parameters. The NME-based surface kinetic models obey physical constraints, which are violated in models based on neural ordinary differential equations. The proposed NME framework for multiscale modeling of molecular processes can pave the way for the discovery of new chemistries and materials in atomic-scale plasma processes.

Chemical engineering↗
DOE OSTI2024

Characterizing gravitational wave detector networks: from A # to cosmic explorer

Gravitational-wave observations by the laser interferometer gravitational-wave observatory (LIGO) and Virgo have provided us a new tool to explore the Universe on all scales from nuclear physics to the cosmos and have the massive potential to further impact fundamental physics, astrophysics, and cosmology for decades to come. In this paper we have studied the science capabilities of a network of LIGO detectors when they reach their best possible sensitivity, called A # , given the infrastructure in which they exist and a new generation of observatories that are factor of 10 to 100 times more sensitive (depending on the frequency), in particular a pair of L-shaped cosmic explorer (CE) observatories (one 40 km and one 20 km arm length) in the US and the triangular Einstein telescope with 10 km arms in Europe. We use a set of science metrics derived from the top priorities of several funding agencies to characterize the science capabilities of different networks. The presence of one or two A # observatories in a network containing two or one next generation observatories, respectively, will provide good localization capabilities for facilitating multimessenger astronomy (MMA) and precision measurement of the Hubble parameter. Two CE observatories are indispensable for achieving precise localization of binary neutron star events, facilitating detection of electromagnetic counterparts and transforming MMA. Their combined operation is even more important in the detection and localization of high-redshift sources, such as binary neutron stars, beyond the star-formation peak, and primordial black hole mergers, which may occur roughly 100 million years after the Big Bang. The addition of the Einstein Telescope to a network of two CE observatories is critical for accomplishing all the identified science metrics including the nuclear equation of state, cosmological parameters, the growth of black holes through cosmic history, but also make new discoveries such as the presence of dark matter within or around neutron stars and black holes, continuous gravitational waves from rotating neutron stars, transient signals from supernovae, and the production of stellar-mass black holes in the early Universe. For most metrics the triple network of next generation terrestrial observatories are a factor 100 better than what can be accomplished by a network of three A # observatories.

Einstein telescope↗
DOE OSTI2026

Spatiotemporal control of laser intensity using differentiable programming

Optical techniques for spatiotemporal control can produce laser pulses with custom amplitude, phase, or polarization structure. In nonlinear optics and plasma physics, the use of structured pulses typically follows a forward design approach, in which the efficacy of a known structure is analyzed for a particular application. Inverse approaches, in contrast, enable the discovery of new structures with the potential for superior performance. Here, an implementation of the unidirectional pulse propagation equation that supports automatic differentiation is combined with gradient-based optimization to design structured pulses with features that are advantageous for a range of nonlinear optical and plasma-based applications: (1) a longitudinally uniform intensity over an extended region, (2) a superluminal intensity peak that travels many Rayleigh ranges with constant duration, spot size, and amplitude, and (3) a laser pulse that ionizes a gas to form a uniform column of plasma. In the final case, optimizing the full spatiotemporal structure improves the performance by a factor of 15 compared to optimizing only spatial or only temporal structure, highlighting the advantage of spatiotemporal control.

automatic differentiation↗
DOE OSTI2026

Neural entropy-stable conservative flux form neural networks for learning hyperbolic conservation laws

We propose a neural entropy-stable conservative flux form neural network (NESCFN) for learning hyperbolic conservation laws and their associated entropy functions directly from solution trajectories, without requiring any predefined numerical discretization. While recent neural network architectures have successfully integrated classical numerical principles into learned models, most rely on prior knowledge of the governing equations or assume a fixed discretization. Our approach removes this dependency by embedding entropy-stable design principles into the learning process itself, enabling the discovery of physically consistent dynamics in a fully data-driven setting. By jointly learning both the flux function and a corresponding entropy, NESCFN promotes conservation and entropy dissipation, which is critical for long-term stability and fidelity in the system of hyperbolic conservation laws. Furthermore, numerical results demonstrate that the method achieves stability and conservation over extended time horizons and accurately captures shock propagation speeds, even without oracle access to future-time solution profiles in the training data.

Conservative flux form↗
DOE OSTI2026

Scientific Discovery with Physics-Informed System Identification (Abbreviated Report)

My fellowship research focused on making physics-based simulations faster and more useful through machine learning. Many problems in science and engineering are governed by partial differential equations, but high-fidelity simulations are often too expensive to run repeatedly. I worked on improving Latent Space Dynamics Identification (LaSDI), a reduced-order modeling framework that compresses large simulation data sets into a smaller representation and then learns how that representation evolves over time. The motivation was to develop reduced models that remain accurate for more challenging systems, especially when predictions must remain reliable over long time intervals or when the underlying dynamics are more complicated than standard methods can easily handle. I also contributed to related work on Quandary, a high-performance software effort for simulation and control of open quantum systems, before focusing primarily on Latent Space Dynamics Identification methods. The main outcomes of the fellowship were two new algorithms (both of which were published), Rollout-LaSDI and Higher-Order LaSDI, together with supporting work on multi-stage Latent Space Dynamics Identification. Rollout-LaSDI improved long-term prediction by training the model to stay accurate over extended time horizons, and Higher-Order LaSDI broadened the method so it could model systems with higher-order time dynamics. My contributions to multistage Latent Space Dynamics Identification also helped show that its later training stages could be simplified without losing effectiveness, and that this behavior held across different model architectures and training strategies. Taken together, these advances improved the accuracy, flexibility, and practical value of reduced-order modeling tools for computational science.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Differential equations for cosmological correlators

Cosmological fluctuations retain a memory of the physics that generated them in their spatial correlations. The strength of correlations varies smoothly as a function of external kinematics, which is encoded in differential equations satisfied by cosmological correlation functions. In this work, we provide a broader perspective on the origin and structure of these differential equations. As a concrete example, we study conformally coupled scalar fields in a power-law cosmology. The wavefunction coefficients in this model have integral representations, with the integrands being the product of the corresponding flat-space results and “twist factors” that depend on the cosmological evolution. Similar twisted integrals arise for loop amplitudes in dimensional regularization, and their recent study has led to the discovery of rich mathematical structures and powerful new tools for computing multi-loop Feynman integrals in quantum field theory. The integrals of interest in cosmology are also part of a finite-dimensional basis of master integrals, which satisfy a system of first-order differential equations. We develop a formalism to derive these differential equations for arbitrary tree graphs. The results can be represented in graphical form by associating the singularities of the differential equations with a set of graph tubings. Upon differentiation, these tubings grow in a local and predictive fashion. In fact, a few remarkably simple rules allow us to predict — by hand — the equations for all tree graphs. While the rules of this “kinematic flow” are defined purely in terms of data on the boundary of the spacetime, they reflect the physics of bulk time evolution. We also study the analogous structures in tr ϕ 3 theory, and see some glimpses of hidden structure in the sum over planar graphs. This suggests that there is an autonomous combinatorial or geometric construction from which cosmological correlations, and the associated spacetime, emerge.

Cosmological models↗
DOE OSTI2026

Semi-Analytical Hierarchical Bayesian Inference of Nonlinear Model Structure in Stochastic Dynamics: Applied to Compartmental Models of Infectious Diseases

A Bayesian computational framework for parsimonious inference in stochastic nonlinear dynamical systems is presented. This framework enables the concurrent estimation of system states, time-varying parameters, time-invariant parameters, and the optimal sparsity structure of the model parameters. Because differential equation-based models are often simplified mechanistic or phenomenological representations, robust inference from noisy measurement data requires explicit treatment of model error and uncertainty. Model error and time-varying parameters can be represented as random processes, enabling inference while making minimal assumptions about the underlying sources of discrepancy and variability. Adopting stochastic differential equation representations affords the model significant flexibility, but can also render it susceptible to overfitting during statistical inversion, where the inferred model may track noise rather than the underlying signal. To alleviate the effects of overfitting and to enable the discovery of the optimal sparse representation of the time-invariant parameters, a Bayesian sparse learning algorithm is embedded within the framework. This sparse learning framework adopts an approximate hierarchical Bayesian setting defined by a series of semi-analytical expressions. The model structure inference framework is validated using a stochastic compartmental model for tracking and forecasting active cases of an infectious disease. Compartmental models describe population-level infectious disease dynamics through interactions among population fractions grouped by disease state. Mathematically, such models consist of a system of coupled ordinary differential equations. This example adopts an expressive compartmental model that includes multiple possible interactions between disease states, motivated by early uncertainty surrounding COVID-19 reinfection dynamics and their implications for long-term epidemic forecasting. The sparse learning exercise permits the inference of a priori unknown epidemiological dynamics from simulated public health data, discovering the nested compartmental model that optimizes the trade-off between average data-fit and model complexity. It is shown that inducing sparsity among the model parameters eliminates redundant interactions between compartments, equivalently revealing the optimal coupling structure between differential equations.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Short-Wave Infrared Upconverting Nanoparticles

Optical technologies enable real-time, noninvasive analysis of complex systems but are limited to discrete regions of the optical spectrum. While wavelengths in the short-wave infrared (SWIR) window (typically, 1700-3000 nm) should enable deep subsurface penetration and reduced photodamage, there are few luminescent probes that can be excited in this region. Here, we report the discovery of lanthanide-based upconverting nanoparticles (UCNPs) that efficiently convert 1740 or 1950 nm excitation to wavelengths compatible with conventional silicon detectors. Screening of Ln3+ ion combinations by differential rate equation modeling identifies Ho3+/Tm3+ or Tm3+ dopants with strong visible or NIR-I emission following SWIR excitation. Experimental upconverted photoluminescence excitation (U-PLE) spectra find that 10% Tm3+-doped NaYF4 core/shell UCNPs have the strongest 800 nm emission from SWIR wavelengths, while UCNPs with an added 2% or 10% Ho3+ show the strongest red emission when excited at 1740 or 1950 nm. Mechanistic modeling shows that addition of a low percentage of Ho3+ to Tm3+-doped UCNPs shifts their emission from 800 to 652 nm by acting as a hub of efficient SWIR energy acceptance and redistribution up to visible emission manifolds. Parallel experimental and computational analysis shows rate equation models are able to predict compositions for specific wavelengths of both excitation and emission. These SWIR-responsive probes open a new IR bioimaging window, and are responsive at wavelengths important for vision technologies.

Qi, Xiao↗