Complex Physical Systems Research Articles

Data-driven inference of high-dimensional spatiotemporal state of plasma systems

Many plasma systems and technologies, such as Hall thrusters for spacecraft propulsion, exhibit complex underlying physics that affect the global operation. When characterizing such systems in an experiment, obtaining full spatiotemporal maps of the involved state variables can be, thus, highly informative. However, this goal is not practically realizable because of various experimental limitations, e.g., finite spatial resolution of the diagnostics and geometrical accessibility constraints. Therefore, having the capability to reconstruct the full high-dimensional states of plasma systems from low-dimensional time-history measurements is greatly desirable. Compressed sensing is a signal processing technique that can answer this crucial need. However, existing compressed sensing approaches have several limitations that restrict their effectiveness for complex physical systems like plasma technologies. These include the need for abundant sensor measurements and a principled sensor placement. In this paper, we demonstrate the capabilities of Shallow Recurrent Decoder (SHRED) architecture for compressed sensing. We show in several plasma test cases that SHRED can robustly infer full high-dimensional spatiotemporal state vectors of these systems (i.e., all macroscopic plasma properties) from minimal system information. This minimal information can consist of three finite time-history measurements of either local values of a plasma property or the global plasma properties (spatially averaged or performance parameters). An application of SHRED's inference capability in the numerical plasma simulation context is “super-resolution” enhancement. We will discuss this application by presenting how SHRED can effectively establish mappings between a low-resolution and a high-resolution simulation, recovering detailed spatial plasma features that are below the simulation's grid size.

Digital Twin Generalization with Meta and Geometric Deep Learning

Deep digital twins (DDTs) are deep neural networks that encode the behavior of complex physical systems. DDTs are excellent system representations due to their ability to continuously adapt to operational changes and their capability to capture complex relationships between system components and processes that cannot be explicitly modeled. For this challenge, DDTs benefit greatly from recent success in geometric deep learning (GDL) which allows the integration of information from multiple systems based on schematic representations. A major challenge in training DDTs is their dependence on the quality and representativeness of training data, especially under the dynamic conditions typical in prognostics and health management (PHM). Recent developments in differentiable simulation present new opportunities for optimizing the training data representativeness. In this thesis, we propose a novel meta-learning framework that trains DDTs using the output from differentiable simulators. This setup enables active optimization of training data sampling through gradient computation, enhancing training speed, robustness, and data representativeness. We extend this framework to address challenges in multi-system data integration in power grids and fault detection in railway traction networks. By applying our framework, we aim to tackle significant challenges in forecasting, anomaly detection and sensor-fault analysis using advanced data fusion techniques. Our approach promises substantial improvements in DDT robustness and operational efficiency, with its effectiveness to be demonstrated through empirical studies on both simple and complex case studies within the power systems domain.

Discovering dynamics and parameters of nonlinear oscillatory and chaotic systems from partial observations

Despite rapid progress in data acquisition techniques, many complex physical, chemical, and biological systems remain only partially observable, thus posing the challenge to identify valid theoretical models and estimate their parameters from an incomplete set of experimentally accessible time series. Here, we combine sensitivity methods and ranked-choice model selection to construct an automated hidden dynamics inference framework that can discover predictive nonlinear dynamical models for both observable and latent variables from noise-corrupted incomplete data in oscillatory and chaotic systems. After validating the framework for prototypical FitzHugh-Nagumo oscillations, we demonstrate its applicability to experimental data from squid neuron activity measurements and Belousov-Zhabotinsky reactions, as well as to the Lorenz system in the chaotic regime. Published by the American Physical Society 2024

Nonlinear analysis of compound pendulum model

Abstract This paper investigates the relationships among various nonlinear physi cal equations, with a particular focus on the standard form of the ϕ^4 equation. Based on the theoretical framework of the Jacobi elliptic function, an exact solution for the ϕ^4 equation is derived. A key innovation of this work is the discovery of the consistency between the ϕ^4 equation and the motion equation of the compound pendulum. By utilizing this correspondence, an exact solution for the compound pendulum equation is obtained, grounded in the Jacobi elliptic function theory. Compared to numerical methods, this solution provides higher accuracy and has the potential to be applied to
more complex nonlinear physical systems. This model can also be applied in areas such as vibration damping in building materials, mechanical system analysis, and spacecraft control.

Gesture recognition with Brownian reservoir computing using geometrically confined skyrmion dynamics

Physical reservoir computing leverages the dynamical properties of complex physical systems to process information efficiently, significantly reducing training efforts and energy consumption. Magnetic skyrmions, topological spin textures, are promising candidates for reservoir computing systems due to their enhanced stability, non-linear interactions and low-power manipulation. Traditional spin-based reservoir computing has been limited to quasi-static detection or real-world data must be rescaled to the intrinsic timescale of the reservoir. We address this challenge by time-multiplexed skyrmion reservoir computing, that allows for aligning the reservoir’s intrinsic timescales to real-world temporal patterns. Using millisecond-scale hand gestures recorded with Range-Doppler radar, we feed voltage excitations directly into our device and detect the skyrmion trajectory evolution. This method scales down to the nanometer level and demonstrates competitive or superior performance compared to energy-intensive software-based neural networks. Our hardware approach’s key advantage is its ability to integrate sensor data in real-time without temporal rescaling, enabling numerous applications.

A Transformative Topological Representation for Link Modeling, Prediction and Cross-Domain Network Analysis.

Many complex social, biological, or physical systems are characterized as networks, and recovering the missing links of a network could shed important lights on its structure and dynamics. A good topological representation is crucial to accurate link modeling and prediction, yet how to account for the kaleidoscopic changes in link formation patterns remains a challenge, especially for analysis in cross-domain studies. We propose a new link representation scheme by projecting the local environment of a link into a "dipole plane", where neighboring nodes of the link are positioned via their relative proximity to the two anchors of the link, like a dipole. By doing this, complex and discrete topology arising from link formation is turned to differentiable point-cloud distribution, opening up new possibilities for topological feature-engineering with desired expressiveness, interpretability and generalization. Our approach has comparable or even superior results against state-of-the-art GNNs, meanwhile with a model up to hundreds of times smaller and running much faster. Furthermore, it provides a universal platform to systematically profile, study, and compare link-patterns from miscellaneous real-world networks. This allows building a global link-pattern atlas, based on which we have uncovered interesting common patterns of link formation, i.e., the bridge-style, the radiation-style, and the community-style across a wide collection of networks with highly different nature.

A hybrid of the fractional Vieta–Lucas functions and its application in constrained fractional optimal control systems containing delay

In this investigation, a novel framework is devised to study an important category of fractional-order systems. The fractional Vieta–Lucas functions (FVLFs) and a hybrid of the block-pulse functions (BPFs) with the mentioned functions are introduced. The advantages of the proposed basis are clarified and a novel integral operator is further constructed based on the Riemann–Liouville integral operator. An innovative spectral collocation methodology is introduced. By utilizing the new fractional basis, the principal problem is changed into a simple optimization one containing unspecified parameters. The developed discretization scheme is robust and produces very satisfactory results for constrained complex physical systems containing delays. Four benchmark problems are examined to verify the capability of the method. The experimental outputs certify the exactness and usefulness of the new methodology.

Investigating new solutions for a general form of q$$ \mathfrak{q} $$‐deformed equation: An analytical and numerical study

SummaryThis paper contributes to the study of a new model called the ‐deformed equation or the ‐deformed tanh‐Gordon model. To understand physical systems with violated symmetries. We utilize the ‐expansion approach to solve the ‐deformed equation for specific parameter values. This method generates solutions that provide valuable insights into the system's dynamics and behavior. To verify the accuracy of our solutions, we also apply the finite difference technique to obtain numerical solutions to the ‐deformed equation. This dual approach ensures the reliability of our results. We present our findings using tables and graphics to enhance clarity and facilitate comparison between the analytical and numerical solutions. These visual aids help readers better understand the similarities and differences between the two approaches. The ‐deformation is significant as it models physical systems with nonstandard symmetry features, like extensivity, offering a more accurate representation of real‐world phenomena. The growing significance of this equation across various disciplines highlights its potential in advancing our understanding of complex physical systems. This paper contributes valuable knowledge about the ‐deformed equation, demonstrating its potential for accurately modeling physical systems with violated symmetries. Through both analytical and numerical techniques, we offer comprehensive solutions and validate their accuracy, with graphical representations enhancing the clarity and understanding of our results. This exploration of ‐deformation advances modeling techniques, providing a more precise depiction of real‐world processes with nonstandard symmetry features.

Neural Network-Based Aggregated Equivalent Modeling of Distributed Photovoltaic External Characteristics of Faults

Distributed power networks have a large number of photovoltaic power sources. The bidirection of power flow, different transient control strategies, and installation locations make the transient characteristics highly complex and unpredictable. The vast network of the distribution system makes it almost impossible to predict the electrical quantities of each branch. Reasonable aggregation modeling of the distribution network can greatly simplify the network topology, facilitating transient control and the setting of relay protection settings. An aggregated equivalent modeling method based on the LSTM neural network for distributed PV fault external characteristics is proposed. This method equates the complex distribution network to a highly nonlinear but controllable current source. The method can output the I–V curves of equivalent PV system parallel points under any output power and is able to predict the fault characteristics of the equivalent system after a voltage drop at the parallel point. Compared to traditional mechanistic modeling, this method does not require specific modeling of complex physical systems and is able to accurately map the strong nonlinear inputs and outputs of distribution networks. The established LSTM model first uses a one-dimensional convolutional layer for feature extraction of the PV power coefficients (input), and then two hidden layers are utilized to process the sequence data; the vectors are mapped into a sequence of external characteristic curves (output) in a fully connected layer. A typical distribution network is built based on the traditional PV power model, and a large number of different output combinations are selected for simulation to provide an effective training set and validation set data for LSTM model training. By using the training set data, the weights and offset coefficients of each layer of the LSTM are continuously optimized until the model with the smallest overall error is obtained, which is the optimal model. Finally, the optimal model is utilized to establish an equivalent distribution network system, different degrees of voltage drops are set up at the grid-connected points, the fault characteristics are compared with those of the complete model, and the simulation results can prove the reliability and practicality of the proposed method.

Zonewise surrogate-based optimization of box-constrained systems

Complex physical or numerical systems may exhibit distinct behaviors in various zones of their design spaces. We present an algorithm that uses multiple cluster-based surrogates for optimizing such box-constrained systems. It partitions the design space into multiple clusters using K-means clustering and develops a separate surrogate for each cluster. It then uses these surrogates to sample additional points in the design space whose function evaluations guide the search for a global optimum. Clustering, surrogate construction, and smart sampling are employed iteratively to add sample points until a pre-defined threshold. The best solution from these points estimates a global optimum. An extensive test bed of 52 box-constrained functions was used to evaluate and compare the algorithm's performance and computational requirements with sixteen derivative-free optimization solvers. The best version of our algorithm surpassed all sixteen solvers in optimization accuracy for a fixed number of evaluations and demanded lower computational effort than fifteen.

Advanced neural network approaches for coupled equations with fractional derivatives

We investigate numerical solutions and compare them with Fractional Physics-Informed Neural Network (FPINN) solutions for a coupled wave equation involving fractional partial derivatives. The problem explores the evolution of functions u and v over time t and space x. We employ two numerical approximation schemes based on the finite element method to discretize the system of equations. The effectiveness of these schemes is validated by comparing numerical results with exact solutions. Additionally, we introduce the FPINN method to tackle the coupled equation with fractional derivative orders and compare its performance against traditional numerical methods.Key findings reveal that both numerical approaches provide accurate solutions, with the FPINN method demonstrating competitive performance in terms of accuracy and computational efficiency. Our study highlights the significance of employing FPINNs in solving fractional differential equations and underscores their potential as alternatives to conventional numerical methods. The novelty of this work lies in its comparative analysis of traditional numerical techniques and FPINNs for solving coupled wave equations with fractional derivatives, offering insights into advancing computational methods for complex physical systems.

Reducing the number of qubits in quantum simulations of one dimensional many-body Hamiltonians

We investigate the Ising and Heisenberg models using the block renormalization group method (BRGM), focusing on its behavior across different system sizes. The BRGM reduces the number of spins by a factor of 1/2 (1/3) for the Ising (Heisenberg) model, effectively preserving essential physical features of the model while using only a fraction of the spins. Through a comparative analysis, we demonstrate that as the system size increases, there is an exponential convergence between results obtained from the original and renormalized Ising Hamiltonians, provided the coupling constants are redefined accordingly. Remarkably, for a spin chain with 24 spins, all physical features, including magnetization, correlation function, and entanglement entropy, exhibit an exact correspondence with the results from the original Hamiltonian. The study of the Heisenberg model also shows this tendency, although complete convergence may appear for a size much larger than 24 spins, and is therefore beyond our computational capabilities. The success of BRGM in accurately characterizing the Ising model, even with a relatively small number of spins, underscores its robustness and utility in studying complex physical systems, and facilitates its simulation on current NISQ computers, where the available number of qubits is largely constrained.

Artificial Neural Networks as Efficient Models of Proton Exchange Membrane Fuel Cells

Abstract Utilization of machine learning methodologies, particularly artificial neural networks (ANNs), presents a good approach to accurately model physical systems. Such predictive simulations offer the ability to predict system performance across diverse operational conditions without the need of use mathematical descriptions. This approach contrast with traditional, time-consuming and unstable approaches reliant on partial differential equations. In this study, ANN methodology is used to obtain the characteristics of proton exchange membrane fuel cells (PEMFCs). PEMFC are low temperature devices which convert chemical energy into electricity. This devices are promising applications in the automotive sector. Utilizing data gained from computational fluid dynamics simulations of PEMFCs, was explored various data collection techniques and network architectures to check their impact on predictive fidelity. Conclusions show that the ANN-based framework enables rapid prediction of current-voltage characteristics, achieving accuracy levels surpassing 90%. Practical of machine learning model implications were discussed, accenting its utility in optimizing PEMFC operational parameters and its potential integration within digital twin frameworks as a data-driven surrogate model. This study underscores the efficiency of machine learning techniques in advancing the comprehension and optimization of complex physical systems such as PEM-FCs, thereby paving the way for their use in engineering and energy sectors.

Machine-learning-coined noise induces energy-saving synchrony.

Noise-induced synchronization is a pervasive phenomenon observed in a multitude of natural and engineering systems. Here, we devise a machine learning framework with the aim of devising noise controllers to achieve synchronization in diverse complex physical systems. We find the implicit energy regularization phenomenon of the formulated framework that engenders energy-saving artificial noise and we rigorously elucidate the underlying mechanism driving this phenomenon. We substantiate the practical feasibility and efficacy of this framework by testing it across various representative systems of physical and biological significance, each influenced by distinct constraints reflecting real-world scenarios.

Computing macroscopic reaction rates in reaction-diffusion systems using Monte Carlo simulations.

Stochastic reaction-diffusion models are employed to represent many complex physical, biological, societal, and ecological systems. The macroscopic reaction rates describing the large-scale, long-time kinetics in such systems are effective, scale-dependent renormalized parameters that need to be either measured experimentally or computed by means of a microscopic model. In a Monte Carlo simulation of stochastic reaction-diffusion systems, microscopic probabilities for specific events to happen serve as the input control parameters. To match the results of any computer simulation to observations or experiments carried out on the macroscale, a mapping is required between the microscopic probabilities that define the Monte Carlo algorithm and the macroscopic reaction rates that are experimentally measured. Finding the functional dependence of emergent macroscopic rates on the microscopic probabilities (subject to specific rules of interaction) is a very difficult problem, and there is currently no systematic, accurate analytical way to achieve this goal. Therefore, we introduce a straightforward numerical method of using lattice Monte Carlo simulations to evaluate the macroscopic reaction rates by directly obtaining the count statistics of how many events occur per simulation time step. Our technique is first tested on well-understood fundamental examples, namely, restricted birth processes, diffusion-limited two-particle coagulation, and two-species pair annihilation kinetics. Next we utilize the thus gained experience to investigate how the microscopic algorithmic probabilities become coarse-grained into effective macroscopic rates in more complex model systems such as the Lotka-Volterra model for predator-prey competition and coexistence, as well as the rock-paper-scissors or cyclic Lotka-Volterra model and its May-Leonard variant that capture population dynamics with cyclic dominance motifs. Thereby we achieve a more thorough and deeper understanding of coarse graining in spatially extended stochastic reaction-diffusion systems and the nontrivial relationships between the associated microscopic and macroscopic model parameters, with a focus on ecological systems. The proposed technique should generally provide a useful means to better fit Monte Carlo simulation results to experimental or observational data.

Physics-informed neural networks with hard linear equality constraints

Surrogate modeling is used to replace computationally expensive simulations. Neural networks have been widely applied as surrogate models that enable efficient evaluations over complex physical systems. Despite this, neural networks are data-driven models and devoid of any physics. The incorporation of physics into neural networks can improve generalization and data efficiency. The physics-informed neural network (PINN) is an approach to leverage known physical constraints present in the data, but it cannot strictly satisfy them in the predictions. This work proposes a novel physics-informed neural network, KKT-hPINN, which rigorously guarantees hard linear equality constraints through projection layers derived from KKT conditions. Numerical experiments on Aspen models of a continuous stirred-tank reactor (CSTR) unit, an extractive distillation subsystem, and a chemical plant demonstrate that this model can further enhance the prediction accuracy.

Comparison of unsteady MHD flow of second grade fluid by two fractional derivatives

A study is conducted on the unsteady motion of a free convective flow of second grade fluid, energy transfer, and Darcy’s law over an oscillatory smooth vertical plate. The study compares two different approaches in developing a fractional model: the Caputo-Fabrizio operator with nonsingular kernel and the constant proportional Caputo fractional operator with Fourier’s and Fick’s laws. By applying the Laplace method and transforming the provided set of equations into nondimensional form, we obtained semi-analytical results and presented these results through graphical analysis. The study examines how different flow parameters, including the fractional parameters, affect the velocity, mass, and heat profiles of the physical system. The results suggest that the physical model using constant proportional Caputo derivative leads to higher temperature, stronger concentration, and increased velocity compared to the Caputo-Fabrizio model. This highlights the importance of selecting an appropriate fractional model when studying complex physical systems. It is also depicted from the whole analysis that field variables with novel hybrid fractional derivative constant proportional Caputo exhibits a more effective and declining tendency than Caputo-Fabrizio.

Classifying topology in photonic crystal slabs with radiative environments

In the recent years, photonic Chern materials have attracted substantial interest as they feature topological edge states that are robust against disorder, promising to realize defect-agnostic integrated photonic crystal slab devices. However, the out-of-plane radiative losses in those photonic Chern slabs has been previously neglected, yielding limited accuracy for predictions of these systems’ topological protection. Here, we develop a general framework for measuring the topological protection in photonic systems, such as in photonic crystal slabs, while accounting for in-plane and out-of-plane radiative losses. Our approach relies on the spectral localizer that combines the position and Hamiltonian matrices of the system to draw a real-picture of the system’s topology. This operator-based approach to topology allows us to use an effective Hamiltonian directly derived from the full-wave Maxwell equations after discretization via finite-elements method (FEM), resulting in the full account of all the system’s physical processes. As the spectral FEM-localizer is constructed solely from FEM discretization of the system’s master equation, the proposed framework is applicable to any physical system and is compatible with commonly used FEM software. Moving forward, we anticipate the generality of the method to aid in the topological classification of a broad range of complex physical systems.

Data-driven identification of stable sparse differential operators using constrained regression

Identifying differential operators from data is essential for the mathematical modeling of complex physical and biological systems where massive datasets are available. These operators must be stable for accurate predictions for dynamics forecasting problems. In this article, we propose a novel methodology for learning differential operators that are theoretically linearly stable and have sparsity patterns of common discretization schemes. These differential operators are obtained by solving a constrained regression problem, involving local constraints to ensure the linear stability of the global dynamical system. We further extend this approach for learning nonlinear differential operators by determining linear stability constraints for linearized equations around an equilibrium point. The applicability of the proposed method is demonstrated for both linear and nonlinear partial differential equations such as 1-D scalar advection-diffusion equation, 1-D Burgers equation and 2-D advection equation. The results indicated that solutions to constrained regression problems with linear stability constraints provide accurate and linearly stable sparse differential operators.

Lie symmetry analysis and solitary wave solution of biofilm model Allen-Cahn

The investigation presented in this study delves into the analysis of Lie symmetries for the bistable Allen-Cahn (BAC) equation with a quartic potential, specifically applied to the biofilm model. By employing the Lie symmetry method, we have acquired the Lie infinitesimal generators for the considered model. Using a transformation method, the nonlinear partial differential equations (NPDEs) are converted into various nonlinear ordinary differential equations (NLODEs), providing the numerous closed-form solitary wave solutions. The obtained solutions manifest in various forms including dark, bright, kink, anti-kink, and periodic types using diverse strategies. To enhance the physical interpretation, the study presents 3D, 2D, and contour plots of the acquired solutions. Every graph’s wave-like structure contains information about the structural behaviour of the bacteria that build biofilms on surfaces where rectangles have different densities. This analysis enhances comprehension of the complex dynamics present in areas like fluid dynamics, fiber optics, biology, ocean physics, coastal engineering, and nonlinear complex physical systems.