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DOE OSTI2025

Generative learning for slow manifolds and bifurcation diagrams

In dynamical systems characterized by separation of time scales, the approximation of so called “slow manifolds”, on which the long term dynamics lie, is a useful step for model reduction. Initializing on such slow manifolds is a useful step in modeling, since it circumvents fast transients, and is crucial in multiscale algorithms (like the equation-free approach) alternating between fine scale (fast) and coarser scale (slow) simulations. In a similar spirit, when one studies the infinite time dynamics of systems depending on parameters, the system attractors (e.g., its steady states) lie on bifurcation diagrams (curves for one-parameter continuation, and more generally, on manifolds in state parameter space. Sampling these manifolds gives us representative attractors (here, steady states of ODEs or PDEs) at different parameter values. Algorithms for the systematic construction of these manifolds (slow manifolds, bifurcation diagrams) are required parts of the “traditional” numerical nonlinear dynamics toolkit. In more recent years, as the field of Machine Learning develops, conditional score-based generative models (cSGMs) have been demonstrated to exhibit remarkable capabilities in generating plausible data from target distributions that are conditioned on some given label. It is tempting to exploit such generative models to produce samples of data distributions (points on a slow manifold, steady states on a bifurcation surface) conditioned on (consistent with) some quantity of interest (QoI, observable). In this work, we present a framework for using cSGMs to quickly (a) initialize on a low-dimensional (reduced-order) slow manifold of a multi-time-scale system consistent with desired value(s) of a QoI (a “label”) on the manifold, and (b) approximate steady states in a bifurcation diagram consistent with a (new, out-of-sample) parameter value. This conditional sampling can help uncover the geometry of the reduced slow-manifold and/or approximately “fill in” missing segments of steady states in a bifurcation diagram. Finally, the quantity of interest, which determines how the sampling is conditioned, is either known a priori or identified using manifold learning-based dimensionality reduction techniques applied to the training data.

Dynamical systems↗

DOE OSTI2025

Low‐dimensional manifold learning for uncertainty quantification in complex multi‐scale stochastic systems

Broadly speaking, the goals of the project are to develop techniques to use manifold learning to develop reduced‐order and surrogate models for "hyper‐reduction" of very high‐dimensional complex multi‐scale systems. This is being achieved by employing a newly proposed form of manifold projection and learning that leverages recent advancements in computational geometry and data‐driven modeling. In particular, we are applying a manifold projection technique to project the solutions of very high‐dimensional systems onto the so‐called Grassmannmanifold, a Reimannian manifold comprised of orthonormal matrices. We then apply data‐driven machine learning techniques to classify the solutions on the manifold (e.g. clustering techniques) according to their proximity on the manifold and leverage a further nonlinear dimension reduction to organize the structured data on the manifold. Finally, we are developing novel techniques that enable us to directly interpolate the hyper‐reduced data such that we can predict the solution of the complex, high‐ dimensional system without need to call the full expensive computational model. Given their adherence to the underlying structure of the solution of the physical system, it is expected that these approximate solutions will be sufficiently constrained so as to (approximately) adhere to physical principles.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

Tracking the topology of neural manifolds across populations

Neural manifolds summarize the intrinsic structure of the information encoded by a population of neurons. Advances in experimental techniques have made simultaneous recordings from multiple brain regions increasingly commonplace, raising the possibility of studying how these manifolds relate across populations. However, when the manifolds are nonlinear and possibly code for multiple unknown variables, it is challenging to extract robust and falsifiable information about their relationships. We introduce a framework, called the method of analogous cycles, for matching topological features of neural manifolds using only observed dissimilarity matrices within and between neural populations. We demonstrate via analysis of simulations and in vivo experimental data that this method can be used to correctly identify multiple shared circular coordinate systems across both stimuli and inferred neural manifolds. Conversely, the method rejects matching features that are not intrinsic to one of the systems. Further, as this method is deterministic and does not rely on dimensionality reduction or optimization methods, it is amenable to direct mathematical investigation and interpretation in terms of the underlying neural activity. We thus propose the method of analogous cycles as a suitable foundation for a theory of cross-population analysis via neural manifolds.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

Porous mesh manifold for enhanced boiling performance

High-performance electronics are continuously demanding cooling of higher heat fluxes. Phase-change cooling, including pool boiling, is a useful approach to address this challenge; however, competition between liquid and vapor flows generally limit the heat fluxes that can be dissipated. A range of strategies to control these flows have been investigated previously, including capillary guides. Here a manifold structure formed from a metallic mesh is investigated to control the disposition of liquid and vapor phases above a pool fed boiling surface enhanced with porous structures. Copper mesh forms defined liquid flow paths, using capillary action to guide and distribute liquid evenly over the heated surface, along with open channels to facilitate vapor escape. The mesh provides a novel structure for liquid guidance that imposes low resistance to liquid flow while occluding a minimal area of heated surface underneath. The manifold performance is characterized in boiling fed by a pool of water above a laser-textured aluminum nitride heat dissipation surface with pin–fin structures having heights of 110 µm and spacing of 30 µm with a heated area of 5 mm x 5 mm. A maximum heat flux of 490 W/cm 2 is reached with the manifold in the pool fed configuration, representing an increase of more than 65% over the porous pin fin surface alone. The maximum stable superheat observed for the manifold of 36K is 14K higher than that for the porous surface without the manifold. The factors limiting performance of the manifold are analyzed. High superheat is attributed to partial flooding of the boiling surface as suggested by the reduction in superheat using external suction. Similar systems and structures for enhanced two-phase cooling are compared.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2025

Construction of 3D MHD pressure drop correlation and flow characterization in the contraction region of a fusion blanket manifold

Inlet and outlet manifolds are typical components of liquid metal (LM) blanket designs of a fusion power reactor to be used to distribute the LM flow into breeding channels and collect it at the exit of the blanket. High pressure loss in the magnetohydrodynamic (MHD) flows featuring abrupt geometrical changes is one of the main feasibility issues of such designs. Recently, optimization studies were conducted to construct 3D MHD pressure drop correlations for a LM flow in an electrically insulating manifold with gradual expansion. Here, the 3D computational approach developed in that study is applied to the outlet manifold featuring gradual contraction. A systematic analysis was performed with a total number of 135 flow cases computed with COMSOL Multiphysics for Hartmann numbers 1000 < Ha < 10,000, Reynolds numbers 100 < Re < 12,000, and contraction angles 45° < θ < 75° for a fixed contraction ratio of 4. The effects of Ha, Re and θ on the flow recirculation, development length and the total pressure drop were carefully examined. A linear regression analysis was used to determine the power rule of pressure drop coefficient k related to Ha and Re, demonstrating a good match with the Ludford layer theory. Eventually, a correlation for the 3D MHD pressure drop coefficient was constructed as a function of Ha, Re and θ. Further, the results were compared against the inlet manifold. It was found that the flow in the inlet manifold exhibits larger recirculation zones. In the investigated range of Ha, Re and θ, the pressure drop coefficient k of the LM MHD flow in the gradual contraction is only slightly lower (< 8 %) than that in the gradual expansion.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2025

Link Scheduling in Satellite Networks via Machine Learning Over Riemannian Manifolds

Low Earth Orbit (LEO) satellites play a crucial role in enhancing global connectivity, serving a complementary solution to existing terrestrial systems. In wireless networks, scheduling is a vital process that allocates time-frequency resources to users for interference management. However, LEO satellite networks face significant challenges in scheduling their links towards ground users due to the satellites’ mobility and overlapping coverage. This paper addresses the dynamic link scheduling problem in LEO satellite networks by considering spatio-temporal correlations introduced by the satellites’ movements. The first step in the proposed solution involves modeling the network over Riemannian manifolds, thanks to their representation as symmetric positive definite matrices. We introduce two machine learning (ML)-based link scheduling techniques that model the dynamic evolution of satellite positions and link conditions over time and space. To accurately predict satellite link states, we present a recurrent neural network (RNN) over Riemannian manifolds, which captures spatio-temporal characteristics over time. Furthermore, we introduce a separate model, the convolutional neural network (CNN) over Riemannian manifolds, which captures geometric relationships between satellites and users by extracting spatial features from the network topology across all links. Simulation results demonstrate that both RNN and CNN over Riemannian manifolds deliver comparable performance to the fractional programming-based link scheduling (FPLinQ) benchmark. Remarkably, unlike other ML-based models that require extensive training data, both models only need 30 training samples to achieve over 99% of the sum rate while maintaining similar computational complexity relative to the benchmark.

42 ENGINEERING↗

DOE OSTI2025

Neural Active Manifolds: Nonlinear Dimensionality Reduction for Uncertainty Quantification

We present a new approach for nonlinear dimensionality reduction, specifically designed for computationally expensive mathematical models. We leverage autoencoders to discover a one-dimensional neural active manifold (NeurAM) capturing the model output variability, through the aid of a simultaneously learnt surrogate model with inputs on this manifold. Our method only relies on model evaluations and does not require the knowledge of gradients. The proposed dimensionality reduction framework can then be applied to assist outer loop many-query tasks in scientific computing, like sensitivity analysis and multifidelity uncertainty propagation. In particular, we prove, both theoretically under idealized conditions, and numerically in challenging test cases, how NeurAM can be used to obtain multifidelity sampling estimators with reduced variance by sampling the models on the discovered low-dimensional and shared manifold among models. Several numerical examples illustrate the main features of the proposed dimensionality reduction strategy and highlight its advantages with respect to existing approaches in the literature.

Autoencoders↗

DOE OSTI2025

Generative learning of densities on manifolds

A generative modeling framework is proposed that combines diffusion models and manifold learning to efficiently sample data densities on manifolds. The approach utilizes Diffusion Maps to uncover possible low-dimensional underlying (latent) spaces in the high-dimensional data (ambient) space. Two approaches for sampling from the latent data density are described. The first is a score-based diffusion model, which is trained to map a standard normal distribution to the latent data distribution using a neural network. The second one involves solving an Itô stochastic differential equation in the latent space. Additional realizations of the data are generated by lifting the samples back to the ambient space using Double Diffusion Maps , a recently introduced technique typically employed in studying dynamical system reduction; here the focus lies in sampling densities rather than system dynamics. The proposed approaches enable sampling high dimensional data densities restricted to low-dimensional, a priori unknown manifolds. The efficacy of the proposed framework is demonstrated through a benchmark problem and a material with multiscale structure.

Double diffusion maps↗

DOE OSTI2026

A conservative discontinuous Galerkin algorithm for particle kinetics on smooth manifolds

A novel, conservative discontinuous Galerkin algorithm is presented for particle kinetics on manifolds. The motion of particles on the manifold is represented using both canonical and non-canonical Hamiltonian formulations. Our schemes apply to both formulations, but the canonical formulation results in a particularly efficient scheme that also conserves particle density and energy exactly. The collisionless update is coupled to a Bhatnagar-Gross-Krook (BGK) collision operator that provides a simplified model for relaxation to local thermodynamic equilibrium. An iterative scheme is constructed to ensure collisional invariants (density, momentum and energy) are preserved numerically. Rotation of the manifold is incorporated by modifying the Hamiltonian while ensuring a canonical formulation. Several test problems, including a kinetic version of the classical Sod shock problem, Kelvin-Helmholtz instability on the surfaces of a sphere and a hyperboloid, with and without rotations, are presented. A prospectus for further development of this approach to simulation of kinetic theory in general relativity is presented.

Discontinuous Galerkin↗

DOE OSTI2025

Online learning of quadratic manifolds from streaming data for nonlinear dimensionality reduction and nonlinear model reduction

Here, this work introduces an online greedy method for constructing quadratic manifolds from streaming data, designed to enable in situ analysis of numerical simulation data on the Petabyte scale. Unlike traditional batch methods, which require all data to be available upfront and take multiple passes over the data, the proposed online greedy method incrementally updates quadratic manifolds in one pass as data points are received, eliminating the need for expensive disk input/output operations as well as storing and loading data points once they have been processed. A range of numerical examples demonstrate that the online greedy method learns accurate quadratic manifold embeddings while being capable of processing data that far exceed common disk input/output capabilities and volumes as well as main-memory sizes.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

Spectral Bounds on Hyperbolic 3-Manifolds: Associativity and the Trace Formula

We constrain the low-energy spectra of Laplace operators on closed hyperbolic manifolds and orbifolds in three dimensions, including the standard Laplace--Beltrami operator on functions and the Laplacian on powers of the cotangent bundle. Our approach employs linear programming techniques to derive rigorous bounds by leveraging two types of spectral identities. The first type, inspired by the conformal bootstrap, arises from the consistency of the spectral decomposition of the product of Laplace eigensections, and involves the Laplacian spectra as well as integrals of triple products of eigensections. We formulate these conditions in the language of representation theory of PSL 2 (C) and use them to prove upper bounds on the first and second Laplacian eigenvalues. The second type of spectral identities follows from the Selberg trace formula. We use them to find upper bounds on the spectral gap of the Laplace--Beltrami operator on hyperbolic 3-orbifolds, as well as on the systole length of hyperbolic 3-manifolds, as a function of the volume. Further, we prove that the spectral gap λ 1 of the Laplace--Beltrami operator on all closed hyperbolic 3-manifolds satisfies λ 1 < 47.32. Along the way, we use the trace formula to estimate the low-energy spectra of a large set of example orbifolds and compare them with our general bounds, finding that the bounds are nearly sharp in several cases.

Bonifacio, James [University of Mississippi, MS (U↗

DOE OSTI2024

Collocation methods for nonlinear differential equations on low-rank manifolds

We introduce new methods for integrating nonlinear differential equations on low-rank manifolds. These methods rely on interpolatory projections onto the tangent space, enabling low-rank time integration of vector fields that can be evaluated entry-wise. A key advantage of our approach is that it does not require the vector field to exhibit low-rank structure, thereby overcoming significant limitations of traditional dynamical low-rank methods based on orthogonal projection. To construct the interpolatory projectors, we develop a sparse tensor sampling algorithm based on the discrete empirical interpolation method (DEIM) that parameterizes tensor train manifolds and their tangent spaces with cross interpolation. Using these projectors, we propose two time integration schemes on low-rank tensor train manifolds. The first scheme integrates the solution at selected interpolation indices and constructs the solution with cross interpolation. The second scheme generalizes the well-known orthogonal projector-splitting integrator to interpolatory projectors. We demonstrate the proposed methods with applications to several tensor differential equations arising from the discretization of partial differential equations.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2026

Enabling probabilistic learning on manifolds through double diffusion maps

Here, we present a generative learning framework for probabilistic sampling that extends Probabilistic Learning on Manifolds (PLoM), which is designed to generate statistically consistent realizations of a random vector in a finite-dimensional Euclidean space, informed by a (representative) set of observations. In its original form, PLoM constructs a reduced-order probabilistic model by combining three main components: (a) kernel density estimation to approximate the underlying probability measure, (b) Diffusion Maps to characterize the manifold of the data, and (c) a reduced-order Itô Stochastic Differential Equation (ISDE) to sample from the learned distribution. However, its sampling dynamics are posed in the ambient space and the retained number of reduced coordinates is chosen by projection-reconstruction error. In practice, this often (i) requires more coordinates than the data’s intrinsic dimension to achieve stable sampling and (ii) lacks a smooth, basis-independent lifting back to the data domain; moreover, standard Diffusion Maps emphasize harmonic eigenfunctions and can miss non-harmonic latent structure. We address these limitations by decoupling geometry learning from sampling: a first Diffusion Maps pass identifies non-harmonic coordinates on which we formulate a full-order ISDE directly in the latent space, while Double Diffusion Maps captures multiscale geometric features and Geometric Harmonics (GH) learns a smooth lifting map to the ambient variables that is independent of the particular diffusion basis. This hybrid design preserves the system’s dynamical richness with a compact geometric representation and enables principled out-of-sample inference. The effectiveness and robustness of the proposed method are illustrated through two numerical studies: one based on data generated from two-dimensional Hermite polynomial functions and another based on high-fidelity simulations of a detonation wave in a reactive flow.

Double diffusion maps↗

DOE OSTI2025

Generalized fiducial inference on differentiable manifolds

We introduce a novel approach to inference on parameters that take values in a Riemannian manifold embedded in a Euclidean space. Parameter spaces of this form are ubiquitous across many fields, including chemistry, physics, computer graphics, and geology. Here, this new approach uses generalized fiducial inference (GFI) to obtain a posterior-like distribution on the manifold, without needing to know local parameterizations that map to the constrained space from an unconstrained Euclidean space. Using mathematical tools from Riemannian geometry, we construct a constrained generalized fiducial distribution (CGFD). A Bernstein-von Mises-type result for the CGFD, which provides intuition for how the desirable asymptotic qualities of the unconstrained generalized fiducial distribution are inherited by the CGFD, is provided. To illustrate the practical use of the CGFD, we provide a proof-of-concept example in the context of a linear logspline density estimation problem, and demonstrate that CGFD-based confidence sets exhibit desirable coverage properties via simulation. As an application, we fit a CGFD to COVID-19 case count data from North Carolina, USA.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

A Design for Remanufacturing Framework Incorporating Identification, Evaluation, and Validation: A Case Study of Hydraulic Manifold

In recent years, academic researchers and engineers in the industry have widely recognized the necessity of integrating remanufacturing considerations into product design iterations to advance sustainability objectives. Acknowledging the importance of design for remanufacturing (DfRem), efforts were made to develop tools and guidelines that could be implemented in practice. However, such methods largely rely upon experiential insights and qualitative assessments, leaving a gap in the ability to quantitatively assess the economic and environmental impacts of design choices. To bridge this gap, we investigate existing efforts and present a framework for DfRem that integrates established design and remanufacturing practices into a cohesive workflow with quantitative assessments. To demonstrate its efficacy for making practical design changes for remanufacturing, we apply the framework to a hydraulic manifold in a transmission system for heavy-duty tractors. Through this industry-relevant case study, we focus on showcasing the practical utility of our framework. Based on the identified design modifications from remanufacturability analysis, we estimate the reductions in life cycle costs, energy consumption, and emissions. Afterward, the modifications are tested using physical experiments with plans for integration into future iterations of the hydraulic manifold design and production. Here, we anticipate this framework can illustrate the process of remanufacturing that ensures improvements in sustainability while maintaining performance and reliability standards.

design for X↗

DOE OSTI2025

Kernel Manifolds: Nonlinear‐Augmentation Dimensionality Reduction Using Reproducing Kernel Hilbert Spaces

This paper generalizes recent advances on quadratic manifold (QM) dimensionality reduction by developing kernel methods-based nonlinear-augmentation dimensionality reduction. QMs, and more generally feature map-based nonlinear corrections, augment linear dimensionality reduction with a nonlinear correction term in the reconstruction map to overcome approximation accuracy limitations of purely linear approaches. While feature map-based approaches typically learn a least squares optimal polynomial correction term, we generalize this approach by learning an optimal nonlinear correction from a user-defined reproducing kernel Hilbert space. Our approach allows one to impose arbitrary nonlinear structure on the correction term, including polynomial structure, and includes feature map and radial basis function-based corrections as special cases. Furthermore, our method has relatively low training cost and has monotonically decreasing error as the latent space dimension increases. In conclusion, we compare our approach to proper orthogonal decomposition and several recent QM approaches on data from several example problems.

kernel methods↗

DOE OSTI2026

Local reduced-order modeling for electrostatic plasmas by physics-informed solution manifold decomposition

Despite advancements in high-performance computing and modern numerical algorithms, computational cost remains prohibitive for multi-query kinetic plasma simulations. Here, in this work, we develop data-driven reduced-order models (ROMs) for collisionless electrostatic plasma dynamics, based on the kinetic Vlasov-Poisson equation. Our ROM approach projects the equation onto a linear subspace defined by the proper orthogonal decomposition (POD) modes. We introduce an efficient tensorial method to update the nonlinear term using a precomputed third-order tensor. We capture multiscale behavior with a minimal number of POD modes by decomposing the solution manifold into multiple time windows and creating temporally local ROMs. We consider two strategies for decomposition: one based on the physical time and the other based on the electric field energy. Applied to the 1D1V Vlasov–Poisson simulations, that is, prescribed E-field, Landau damping, and two-stream instability, we demonstrate that our ROMs accurately capture the total energy of the system both for parametric and time extrapolation cases. The temporally local ROMs are more efficient and accurate than the single ROM. In addition, in the two-stream instability case, we show that the energy-windowing reduced-order model (EW-ROM) is more efficient and accurate than the time-windowing reduced-order model (TW-ROM). With the tensorial approach, EW-ROM solves the equation approximately 90 times faster than Eulerian simulations while maintaining a maximum relative error of 7.5% for the training data and 11% for the testing data.

Electrostatic plasmas↗

DOE OSTI2025

Conflict Detection in Open RAN with Recurrent Neural Networks Using Geometric Manifolds

Allowing third-party applications on Radio Access Network (RAN) Intelligent Controllers (RICs) within the OpenRAN (O-RAN) framework introduces conflicting interactions that are often difficult to detect in advance. These conflicts, occurring between third-party applications in the Near RealTime RIC (Near-RT RIC), known as xApps, can lead to performance degradation and instability in O-RAN if not identified early. Existing conflict detection and mitigation solutions in the literature assume that the conflicts are known beforehand, which is not always accurate due to the complex and often hidden relationships between control parameters and Key Performance Indicators (KPIs). In this paper, we propose a novel Recurrent Neural Network (RNN) to detect both known and unknown conflicts in O-RAN xApps as specified in the O-RAN standards. We model the xApps, control parameters, and KPIs with nodes and edges to create graph structures and use the hidden nonEuclidean geometric properties of the Riemannian manifold to train the RNN model. The performance of this proposed model is validated using evaluation metrics and compared with benchmarks. Results demonstrate that the proposed RNN model, leveraging Riemannian geometric properties, can achieve 100% of the F1-score provided by an optimal solution in just 20 iterations.

5G↗