High-dimensional Systems Research Articles

Multi-output multi-physics-informed neural network for learning dimension-reduced probability density evolution equation with unknown spatio-temporal-dependent coefficients

The Dimension-Reduced Probability Density Evolution Equation (DR-PDEE) offers a promising approach for evaluating probability density evolution in stochastic dynamical systems. Physics-Informed Neural Networks (PINNs) are well-suited for solving DR-PDEE due to their ability to encode physical laws into the learning process. However, challenges arise from the spatio-temporal-dependence of unknown intrinsic drift and diffusion coefficients, which drive DR-PDEE, along with their derivatives. To address these challenges, a novel framework called Multi-Output Multi-Physics-Informed Neural Network (MO-MPINN) is proposed to predict the evolution of time-varying coefficients and response probability density simultaneously. MO-MPINN features multiple output neurons, eliminating the necessity for distinct identification of unknown spatio-temporal-dependent coefficients separately. It uses parallel subnetworks to reduce training complexity and embeds multiple physical laws in the loss function to ensure an accurate representation of the underlying principles. Leveraging automatic differentiation, MO-MPINN efficiently computes derivatives of coefficients without resorting to numerical differentiation. The framework is applicable to high-dimensional stochastic nonlinear systems with double randomness in structural parameters and excitations. Several structures are presented to validate the performance of the MO-MPINN. This study introduces a new paradigm for solving partial differential equations involving differentiation of spatio-temporal-dependent coefficients.

Local field potential sharp waves with diversified impact on cortical neuronal encoding of haptic input

Cortical sensory processing is greatly impacted by internally generated activity. But controlling for that activity is difficult since the thalamocortical network is a high-dimensional system with rapid state changes. Therefore, to unwind the cortical computational architecture there is a need for physiological ‘landmarks’ that can be used as frames of reference for computational state. Here we use a waveshape transform method to identify conspicuous local field potential sharp waves (LFP-SPWs) in the somatosensory cortex (S1). LFP-SPW events triggered short-lasting but massive neuronal activation in all recorded neurons with a subset of neurons initiating their activation up to 20 ms before the LFP-SPW onset. In contrast, LFP-SPWs differentially impacted the neuronal spike responses to ensuing tactile inputs, depressing the tactile responses in some neurons and enhancing them in others. When LFP-SPWs coactivated with more distant cortical surface (ECoG)-SPWs, suggesting an involvement of these SPWs in global cortical signaling, the impact of the LFP-SPW on the neuronal tactile response could change substantially, including inverting its impact to the opposite. These cortical SPWs shared many signal fingerprint characteristics as reported for hippocampal SPWs and may be a biomarker for a particular type of state change that is possibly shared byboth hippocampus and neocortex.

FPSO/LNG hawser system lifetime assessment by Gaidai multivariate risk assessment method

Floating Production Storage and Offloading (FPSO) unit being an offshore vessel, storing and producing crude oil, prior to crude oil being transported by accompanying shuttle tanker. Critical mooring/hawser strains during offloading operation have to be accurately predicted, in order to maintain operational safety and reliability. During certain types of offloading, excessive hawser tensions may occur, causing operational risks. Current study examines FPSO vessel’s dynamic reactions to hydrodynamic wave-induced loads, given realistic in situ environmental conditions, utilizing the AQWA software package. Current study advocates novel multi-dimensional spatiotemporal risks assessment approach, that is particularly well suited for large dataset analysis, based on numerical simulations (or measurements). Advocated multivariate reliability methodology may be useful for a variety of marine and offshore systems that must endure severe environmental stressors during their intended operational lifespan. Methodology, presented in this study provides advanced capability to efficiently, yet accurately evaluate dynamic system failure, hazard and damage risks, given representative dynamic record of multidimensional system’s inter-correlated critical components. Gaidai risk assessment method being novel dynamic multidimensional system’s lifetime assessment methodology. In order to validate and benchmark Gaidai risk assessment method, in this study it was applied to FPSO and potentially LNG (i.e., Liquid Natural Gas) vessels dynamics. Major advantage of the advocated approach is that there are no existing alternative risk assessment methods, able to tackle unlimited number of system’s dimensions. Accurate multi-dimensional risk assessment had been carried out, based on numerically simulated data, partially verified by available laboratory experiments. Confidence intervals had been given for predicted dynamic high-dimensional system risk levels.

Unscrambling of single-particle wave functions in systems localized through disorder and monitoring

In systems undergoing localization-delocalization quantum phase transitions due to disorder or monitoring, there is a crucial need for robust methods capable of distinguishing phases and uncovering their intrinsic properties. In this paper, we develop a process of finding a Slater determinant representation of free-fermion wave functions that accurately characterizes localized particles, a procedure we dub “unscrambling.” The central idea is to minimize the overlap between envelopes of single-particle wave functions or, equivalently, to maximize the inverse participation ratio of each orbital. This numerically efficient methodology can differentiate between distinct types of wave functions: exponentially localized, power-law localized, and conformal critical, also revealing the underlying physics of these states. The method is readily extendable to systems in higher dimensions. Furthermore, we apply this approach to a more challenging problem involving disordered monitored free fermions in one dimension, where the unscrambling process unveils the presence of a conformal critical phase and a localized area-law quantum Zeno phase. Importantly, our method can also be extended to free fermion systems without particle number conservation, which we demonstrate by estimating the phase diagram of Z2-symmetric disordered monitored free fermions. Our results unlock the potential of utilizing single-particle wave functions to gain valuable insights into the localization transition properties in systems such as monitored free fermions and disordered models. Published by the American Physical Society 2024

Simulations of classical three-body thermalization in one dimension.

One-dimensional systems, such as nanowires or electrons moving along strong magnetic field lines, have peculiar thermalization physics. The binary collision of pointlike particles, typically the dominant process for reaching thermal equilibrium in higher-dimensional systems, cannot thermalize a 1D system. We study how dilute classical 1D gases thermalize through three-body collisions. We consider a system of identical classical point particles with pairwise repulsive inverse power-law potential V_{ij}∝1/|x_{i}-x_{j}|^{n} or the pairwise Lennard-Jones potential. Using Monte Carlo methods, we compute a collision kernel and use it in the Boltzmann equationto evolve a perturbed thermal state with temperature T toward equilibrium. We explain the shape of the kernel and its dependence on the system parameters. Additionally, we implement molecular dynamics simulations of a many-body gas and show agreement with the Boltzmann evolution in the low-density limit. For the inverse power-law potential, the rate of thermalization is proportional to ρ^{2}T^{1/2-1/n}, where ρ is the number density. The corresponding proportionality constant decreases with increasing n.

Projected data assimilation using sliding window proper orthogonal decomposition

Prediction of the state evolution of complex high-dimensional nonlinear systems is challenging due to the nonlinear sensitivity of the evolution to small inaccuracies in the model. Data Assimilation (DA) techniques improve state estimates by combining model simulations with real-time data. Few DA techniques can simultaneously handle nonlinear evolution, non-Gaussian uncertainty, and the high dimension of the state. We recently proposed addressing these challenges using a POD technique that projects the physical and data models into a reduced-dimensional subspace. Proper Orthogonal Decomposition (POD) is a tool to extract spatiotemporal patterns (modes) that dominate the observed data. We combined the POD-based projection operator, computed in an offline fashion, with a DA scheme that models non-Gaussian uncertainty in lower dimensional subspace. If the model parameters change significantly during time evolution, the offline computation of the projection operators ceases to be useful. We address this challenge using a Sliding Window POD (SW-POD), which recomputes the projection operator based on a sliding subset of snapshots from the entire evolution. The physical model projection is updated dynamically in terms of modes and number of modes, and the data model projection is also chosen to promote a sparse approximation. We test the efficacy of this technique on a modified Lorenz'96 model (L96) with a time-varying forcing and compare it with the time-invariant offline projected algorithm. In particular, dynamically determined physical and data model projections decrease the Root Mean Squared Error (RMSE) and the resampling rate.

High-dimensional two-photon quantum controlled phase-flip gate

High-dimensional quantum systems have been used to reveal interesting fundamental physics and to improve information capacity and noise resilience in quantum information processing. However, it remains a significant challenge to realize universal two-photon quantum gates in high dimensions with high success probability. Here, by considering an ion-cavity QED system, we theoretically propose, to the best of our knowledge, the first high-dimensional, deterministic, and universal two-photon quantum gate. By using an optical cavity embedded with a single trapped Ca+40 ion, we achieve a high average fidelity larger than 98% for a quantum controlled phase-flip gate in four-dimensional space, spanned by photonic spin angular momenta and orbital angular momenta. Our proposed system can be an essential building block for high-dimensional quantum information processing, and also provides a platform for studying high-dimensional cavity QED. Published by the American Physical Society 2024

Neural Projection Filter: Learning Unknown Dynamics Driven by Noisy Observations.

In this article, we propose the novel neural stochastic differential equations (SDEs) driven by noisy sequential observations called neural projection filter (NPF) under the continuous state-space models (SSMs) framework. The contributions of this work are both theoretical and algorithmic. On the one hand, we investigate the approximation capacity of the NPF, i.e., the universal approximation theorem for NPF. More explicitly, under some natural assumptions, we prove that the solution of the SDE driven by the semimartingale can be well approximated by the solution of the NPF. In particular, the explicit estimation bound is given. On the other hand, as an important application of this result, we develop a novel data-driven filter based on NPF. Also, under certain condition, we prove the algorithm convergence; i.e., the dynamics of NPF converges to the target dynamics. At last, we systematically compare the NPF with the existing filters. We verify the convergence theorem in linear case and experimentally demonstrate that the NPF outperforms existing filters in nonlinear case with robustness and efficiency. Furthermore, NPF could handle high-dimensional systems in real-time manner, even for the 100-D cubic sensor, while the state-of-the-art (SOTA) filter fails to do it.

Photonic next-generation reservoir computer based on distributed feedback in optical fiber.

Reservoir computing (RC) is a machine learning paradigm that excels at dynamical systems analysis. Photonic RCs, which perform implicit computation through optical interactions, have attracted increasing attention due to their potential for low latency predictions. However, most existing photonic RCs rely on a nonlinear physical cavity to implement system memory, limiting control over the memory structure and requiring long warm-up times to eliminate transients. In this work, we resolve these issues by demonstrating a photonic next-generation reservoir computer (NG-RC) using a fiber optic platform. Our photonic NG-RC eliminates the need for a cavity by generating feature vectors directly from nonlinear combinations of the input data with varying delays. Our approach uses Rayleigh backscattering to produce output feature vectors by an unconventional nonlinearity resulting from coherent, interferometric mixing followed by a quadratic readout. Performing linear optimization on these feature vectors, our photonic NG-RC demonstrates state-of-the-art performance for the observer (cross-prediction) task applied to the Rössler, Lorenz, and Kuramoto-Sivashinsky systems. In contrast to digital NG-RC implementations, we show that it is possible to scale to high-dimensional systems while maintaining low latency and low power consumption.

Construction method and circuit design of a high-dimensional conservative chaotic system with binary linear combinations

Construction method and circuit design of a high-dimensional conservative chaotic system with binary linear combinations

Low-rank multilinear filtering

Linear filtering methods are well-known and have been successfully applied to system identification and equalization problems. However, when high-dimensional systems are modeled, these methods often perform unsatisfactorily due to their slow convergence and to the high number of parameters to estimate, which brings high computational and storage complexities. To cope with these difficulties, the assumption of a low-rank impulse response was recently exploited to derive computationally efficient adaptive tensor filtering methods. However, existing approaches either model the impulse response as a low-rank matrix or a rank-1 tensor. While the former choice can only bring a limited complexity reduction, the latter relies on a strong assumption that is too restrictive for many systems of interest. In this work, we propose an adaptive filtering approach that models the impulse response more generally as a low-rank tensor, with a rank possibly higher than one and order possibly higher than two. This approach is suitable for problems involving the identification of a high-dimensional system whose impulse response has a multilinear low-rank structure. It can overcome the curse of dimensionality by taking advantage of such a structure, which allows breaking a large system identification problem into several smaller ones. Simulation results compare the proposed algorithms with existing ones in terms of their performance, computational complexity, and memory demands. In particular, these results show that when the target impulse response has a tensor structure, our approach can achieve a faster convergence and lower steady-state error than standard LMS while having reduced memory complexity in comparison with the matrix-based approach.

Quench dynamics in lattices above one dimension: The free fermionic case

We begin a systematic investigation of quench dynamics in higher-dimensional lattice systems considering the case of noninteracting fermions with conserved particle number. We prepare the system in a translational-invariant nonequilibrium initial state, the simplest example being a classical configuration with fermions at fixed positions on the lattice, and let it evolve in time. We characterize the system's dynamics by measuring the entanglement between a finite connected region and its complement. We observe the transmutation of entanglement entropy into thermodynamic entropy and investigate how this process depends on the shape and orientation of the region with respect to the underlying lattice. Interestingly, we find that irregular regions display a distinctive multislope entanglement growth, while the dependence on the orientation angle is generically fairly weak. This is particularly true for regions with a large (discrete) rotational symmetry group. The main tool of our analysis is the celebrated quasiparticle picture of Calabrese and Cardy, which we generalize to describe the case at hand. Specifically, we show that for generic initial configurations (even when restricting to classical ones) one has to allow for the production of multiplets involving n>2 quasiparticles and carrying nondiagonal correlations. We obtain quantitatively accurate predictions, tested against exact numerics, and propose an efficient Monte Carlo based scheme to evaluate them for arbitrary connected regions of generic higher-dimensional lattices. Published by the American Physical Society 2024

GIẢM BẬC MÔ HÌNH ĐỘNG HỌC SỬ DỤNG CẮT NGẮN PHƯƠNG THỨC TRONG BÀI TOÁN CHUYỂN ĐỘNG TÒA NHÀ

Modal truncation, an advanced algorithm for model reduction in dynamic systems, efficiently simplifies complex models by selectively discarding less influential eigenmodes, maintaining a balance between computational efficiency and model accuracy. This paper explores the algorithm's application to a 48th order building model. Proceed to reduce this model to lower orders, then analyze errors in time and frequency domains. Modal truncation algorithm systematically reduces model dimensions while preserving critical dynamic attributes. Numerical simulations reveal a favorable reduction order range (from order 6th to order 25th) for optimal balance, with sensitivity observed at order 25th. From the results obtained, depending on specific requirements, users can use a lower-order model corresponding to the allowed error to replace the original system. Recommendations include iterative refinement for adaptive reduction orders and in-depth analysis around critical points. This algorithm becomes an effective method for researchers dealing with high-dimensional dynamic systems, offering simpler yet accurate model representations. As technology develops, continued refinements and applications of modal truncation are expected, solidifying its role in the realm of model reduction.

Solving large-scale variational inequalities with dynamically adjusting initial condition in physics-informed neural networks

This work aims to solve large-scale variational inequalities (VIs), which are equivalent to high-dimensional systems of ordinary differential equations (ODEs). The existing physics-informed neural network (PINN) approach (Wu and Lisser, 2023) shows superior performance for VIs with less than 1000 variables, but fails for VIs of larger size, due to the increasing number of equations and the requirement of an extensive time interval. To overcome this limitation, we present two algorithms that dynamically adjust the initial condition for the PINN. The first algorithm uses multiple PINNs sequentially to decompose the task, where the best prediction from the current PINN serves as the initial condition for the next PINN. The second algorithm uses a single PINN throughout the solution process, immediately taking any improved prediction as an initial condition and refining the PINN to achieve a better prediction. Finally, we demonstrate the effectiveness of the proposed algorithms on a number of large-scale VI problems with up to 100,000 variables.

Dynamic Mode Decomposition With Gaussian Process Regression for Control of High-Dimensional Nonlinear Systems

Abstract In this work, we consider the problem of learning a reduced-order model of a high-dimensional stochastic nonlinear system with control inputs from noisy data. In particular, we develop a hybrid parametric/nonparametric model that learns the “average” linear dynamics in the data using dynamic mode decomposition with control (DMDc) and the nonlinearities and model uncertainties using Gaussian process (GP) regression and compare it with total least-squares dynamic mode decomposition (tlsDMD), extended here to systems with control inputs (tlsDMDc). The proposed approach is also compared with existing methods, such as DMDc-only and GP-only models, in two tasks: controlling the stochastic nonlinear Stuart–Landau equation and predicting the flowfield induced by a jet-like body force field in a turbulent boundary layer using data from large-scale numerical simulations.

The fluctuational transition mechanism of non-hyperbolic chaotic invariant sets

In order to reveal the general escape mechanism of non-hyperbolic chaotic invariant sets under both weak noise limit and finite noise intensity, the simplest example of Hénon map, which represents the stretching and folding of the phase space, is taken to study the escape mechanism under two kinds of global bifurcation: the fractal boundary crisis and the attractor contact crisis. In this paper, we revealed the general exit mechanism by analyzing the escape paths derived from the shooting method and Monte Carlo simulation. Finally, to further demonstrate universality, an example of high-dimensional differential dynamical system, namely the transient chaos Duffing oscillator, is examined, which underscores the main idea of this paper analyzing specific deterministic structures not only enhances the comprehensibility of the escape process, but also allows for predictions of general escape behavior under weak noise intensity.

Robustness of Turing models and gene regulatory networks with a sweet spot.

Traditional linear stability analysis based on matrix diagonalization is a computationally intensive process for high-dimensional systems of differential equations, posing substantial limitations for the exploration of Turing systems of pattern formation where an additional wave-number parameter needs to be investigated. In this paper, we introduce an efficient and intuitive technique that leverages Gershgorin's theorem to determine upper limits on regions of parameter space and the wave number beyond which Turing instabilities cannot occur. This method offers a streamlined avenue for exploring the phase diagrams of other complex multi-parametric models, such as those found in gene regulatory networks in systems biology. Due to its suitability for the asymptotic limit of infinitely large systems, it predicts the existence of a sweet spot in network size for maximal Jacobian stability.

Deep learning probability flows and entropy production rates in active matter

Active matter systems, from self-propelled colloids to motile bacteria, are characterized by the conversion of free energy into useful work at the microscopic scale. They involve physics beyond the reach of equilibrium statistical mechanics, and a persistent challenge has been to understand the nature of their nonequilibrium states. The entropy production rate and the probability current provide quantitative ways to do so by measuring the breakdown of time-reversal symmetry. Yet, their efficient computation has remained elusive, as they depend on the system's unknown and high-dimensional probability density. Here, building upon recent advances in generative modeling, we develop a deep learning framework to estimate the score of this density. We show that the score, together with the microscopic equations of motion, gives access to the entropy production rate, the probability current, and their decomposition into local contributions from individual particles. To represent the score, we introduce a spatially local transformer network architecture that learns high-order interactions between particles while respecting their underlying permutation symmetry. We demonstrate the broad utility and scalability of the method by applying it to several high-dimensional systems of active particles undergoing motility-induced phase separation (MIPS). We show that a single network trained on a system of 4,096 particles at one packing fraction can generalize to other regions of the phase diagram, including to systems with as many as 32,768 particles. We use this observation to quantify the spatial structure of the departure from equilibrium in MIPS as a function of the number of particles and the packing fraction.

Advanced methods for gene network identification and noise decomposition from single-cell data

Central to analyzing noisy gene expression systems is solving the Chemical Master Equation (CME), which characterizes the probability evolution of the reacting species’ copy numbers. Solving CMEs for high-dimensional systems suffers from the curse of dimensionality. Here, we propose a computational method for improved scalability through a divide-and-conquer strategy that optimally decomposes the whole system into a leader system and several conditionally independent follower subsystems. The CME is solved by combining Monte Carlo estimation for the leader system with stochastic filtering procedures for the follower subsystems. We demonstrate this method with high-dimensional numerical examples and apply it to identify a yeast transcription system at the single-cell resolution, leveraging mRNA time-course experimental data. The identification results enable an accurate examination of the heterogeneity in rate parameters among isogenic cells. To validate this result, we develop a noise decomposition technique exploiting time-course data but requiring no supplementary components, e.g., dual-reporters.

Asymmetric controlled remote implementation of operations in different dimensions

Quantum entanglement is essential for the realization of distributed computing and has been implemented in various types of two-level systems. However, high-dimensional quantum protocols are rarely investigated both theoretically and experimentally, despite the remarkable advantages that high-dimensional quantum systems exhibit over two-level systems for certain quantum information and computing tasks. Here, we propose asymmetric protocols that utilize different dimensional systems to prepare the stator and achieve remote state operations with the assistance of shared N-partite graph states. Our protocols demonstrate convincing control power and security, with all implementations achievable through local operations and classical communications. The experimental realization of high-dimensional quantum systems and operations is presented using current technology. Our work contributes towards achieving arbitrary high-dimensional quantum protocols and paves the way for implementing high-dimensional quantum communication and computation.