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Results for “universal approximation theorem”

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All sources NASA NTRS DOE OSTI

DOE OSTI2022

MIONet: Learning Multiple-Input Operators via Tensor Product

As an emerging paradigm in scientific machine learning, neural operators aim to learn operators, via neural networks, that map between infinite-dimensional function spaces. Several neural operators have been recently developed. However, all the existing neural operators are only designed to learn operators defined on a single Banach space; i.e., the input of the operator is a single function. Here, for the first time, we study the operator regression via neural networks for multiple-input operators defined on the product of Banach spaces. We first prove a universal approximation theorem of continuous multiple-input operators. We also provide a detailed theoretical analysis including the approximation error, which provides guidance for the design of the network architecture. Based on our theory and a low-rank approximation, we propose a novel neural operator, MIONet, to learn multiple-input operators. MIONet consists of several branch nets for encoding the input functions and a trunk net for encoding the domain of the output function. Here, we demonstrate that MIONet can learn solution operators involving systems governed by ordinary and partial differential equations. In our computational examples, we also show that we can endow MIONet with prior knowledge of the underlying system, such as linearity and periodicity, to further improve accuracy.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

Learning stochastic dynamics with statistics-informed neural network

We introduce a machine-learning framework named statistics-informed neural network (SINN) for learning stochastic dynamics from data. This new architecture was theoretically inspired by a universal approximation theorem for stochastic systems, which we introduce in this paper, and the projection-operator formalism for stochastic modeling. Here, we devise mechanisms for training the neural network model to reproduce the correct statistical behavior of a target stochastic process. Numerical simulation results demonstrate that a well-trained SINN can reliably approximate both Markovian and non-Markovian stochastic dynamics. We demonstrate the applicability of SINN to coarse-graining problems and the modeling of transition dynamics. Furthermore, we show that the obtained reduced-order model can be trained on temporally coarse-grained data and hence is well suited for rare-event simulations.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1991

Neurocontrol and fuzzy logic: Connections and designs

Artificial neural networks (ANNs) and fuzzy logic are complementary technologies. ANNs extract information from systems to be learned or controlled, while fuzzy techniques mainly use verbal information from experts. Ideally, both sources of information should be combined. For example, one can learn rules in a hybrid fashion, and then calibrate them for better whole-system performance. ANNs offer universal approximation theorems, pedagogical advantages, very high-throughput hardware, and links to neurophysiology. Neurocontrol - the use of ANNs to directly control motors or actuators, etc. - uses five generalized designs, related to control theory, which can work on fuzzy logic systems as well as ANNs. These designs can copy what experts do instead of what they say, learn to track trajectories, generalize adaptive control, and maximize performance or minimize cost over time, even in noisy environments. Design tradeoffs and future directions are discussed throughout.

Werbos, Paul J.↗

DOE OSTI2026

Definitive Assessment of the Accuracy, Variationality, and Convergence of Relativistic Coupled Cluster and Density Matrix Renormalization Group in 100-Orbital Space

Accuracy, variationality, and convergence underpin the reliability of modern electronic structure methods, yet definitive benchmarks in the relativistic regime remain elusive due to the absence of numerically exact full configuration interaction (CI) references. Recent algorithmic advances in the CI framework, enabled by the small-tensor-product (STP) decomposition approach, have dramatically extended the tractable size of the configuration space, making numerically exact CI calculations feasible in large active spaces previously beyond reach. In this paper, we employ the recently developed STP-CI framework to perform large-scale numerically exact CI calculations and directly benchmark relativistic coupled cluster and density matrix renormalization group methods. Definitive benchmarking of approximate relativistic electronic structure methods is ensured through the application of the gap theorem, which provides rigorous error bounds on the CI reference and establishes a controlled standard for assessing accuracy, variationality, and convergence.

Chemical calculations↗

DOE OSTI2023

Quantifying unitary flow efficiency and entanglement for many-body localization

Holography tells us that there should be a mapping between a boundary Hamiltonian and bulk degrees of freedom. We probe the bulk geometry of the Wegner Wilson Flow (WWF) in the context of many-body localization by addressing efficiency and bulk entanglement growth measures through approximating upper bounds on the boundary entanglement entropy. Here, we connect these upper bounds to the Fubini-Study metric and clarify how a central quantity, the information fluctuation complexity, distinguishes bulk unitary rotation from entanglement production. We also give a short new proof of the small incremental entangling theorem in the absence of ancillas, achieving a dimension independent, universal factor of c = 2.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2025

Exotic phases in finite-density ℤ 3 theories

Lattice ℤ 3 theories with complex actions share many key features with finite- density QCD including a sign problem and $\mathcal{CK}$ symmetry. Complex ℤ 3 spin and gauge models exhibit a generalized Kramers-Wannier duality mapping them onto chiral ℤ 3 spin and gauge models, which are simulatable with standard lattice methods in large regions of parameter space. The Migdal-Kadanoff real-space renormalization group (RG) preserves this duality, and we use it to compute the approximate phase diagram of both spin and gauge ℤ 3 models in dimensions one through four. Chiral ℤ 3 spin models are known to exhibit a Devil’s Flower phase structure, with inhomogeneous phases that can be thought of as ℤ 3 analogues of chiral spirals. Out of the large class of models we study, we find that only chiral spin models and their duals have a Devil’s Flower structure with an infinite set of inhomogeneous phases, a result we attribute to Elitzur’s theorem. We also find that different forms of the Migdal-Kadanoff RG produce different numbers of phases, a violation of the expectation for universal behavior from a real-space RG. We discuss extensions of our work to ℤ N models, SU(N) models and nonzero temperature.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2025

QCD factorization with multihadron fragmentation functions

Important aspects of quantum chromodynamics (QCD) factorization theorems are the properties of the objects involved that can be identified as universal. One example is that the definitions of parton densities and fragmentation functions for different types of hadrons differ only in the identity of the nonperturbative states that form the matrix elements, but are otherwise the same. This leads to independence of perturbative calculations on nonperturbative details of external states. It also lends support to interpretations of correlation functions as encapsulations of intrinsic nonperturbative properties. These characteristics have usually been presumed to still hold true in fragmentation functions even when the observed nonperturbative state is a small-mass cluster of n hadrons rather than simply a single isolated hadron. However, the multidifferential aspect of cross sections that rely on these latter types of fragmentation functions complicates the treatment of kinematical approximations in factorization derivations. That has led to recent claims that the operator definitions for fragmentation functions need to be modified from the single hadron case with nonuniversal prefactors. With such concerns as our motivation, we retrace the steps for factorizing the unpolarized semi-inclusive e + e − annihilation cross section and confirm that they do apply without modification to the case of a small-mass multihadron observed in the final state. In particular, we verify that the standard operator definition from single hadron fragmentation, with its usual prefactor, remains equally valid for the small-mass n -hadron case with the same hard parts and evolution kernels, whereas the more recently proposed definitions with nonuniversal prefactors do not. Our results reaffirm the reliability of most past phenomenological applications of dihadron fragmentation functions. Published by the American Physical Society 2025

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

NASA NTRS1995

Dissipative rendering and neural network control system design

Model-based control system designs are limited by the accuracy of the models of the plant, plant uncertainty, and exogenous signals. Although better models can be obtained with system identification, the models and control designs still have limitations. One approach to reduce the dependency on particular models is to design a set of compensators that will guarantee robust stability to a set of plants. Optimization over the compensator parameters can then be used to get the desired performance. Conservativeness of this approach can be reduced by integrating fundamental properties of the plant models. This is the approach of dissipative control design. Dissipative control designs are based on several variations of the Passivity Theorem, which have been proven for nonlinear/linear and continuous-time/discrete-time systems. These theorems depend not on a specific model of a plant, but on its general dissipative properties. Dissipative control design has found wide applicability in flexible space structures and robotic systems that can be configured to be dissipative. Currently, there is ongoing research to improve the performance of dissipative control designs. For aircraft systems that are not dissipative active control may be used to make them dissipative and then a dissipative control design technique can be used. It is also possible that rendering a system dissipative and dissipative control design may be combined into one step. Furthermore, the transformation of a non-dissipative system to dissipative can be done robustly. One sequential design procedure for finite dimensional linear time-invariant systems has been developed. For nonlinear plants that cannot be controlled adequately with a single linear controller, model-based techniques have additional problems. Nonlinear system identification is still a research topic. Lacking analytical models for model-based design, artificial neural network algorithms have recently received considerable attention. Using their universal approximation property, neural networks have been introduced into nonlinear control designs in several ways. Unfortunately, little work has appeared that analyzes neural network control systems and establishes margins for stability and performance. One approach for this analysis is to set up neural network control systems in the framework presented above. For example, one neural network could be used to render a system to be dissipative, a second strictly dissipative neural network controller could be used to guarantee robust stability.

Gonzalez, Oscar R.↗