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Identifying stochastic dynamics via finite expression methods
Modeling stochastic differential equations (SDEs) is crucial for understanding complex dynamical systems in various scientific fields. Recent methods often employ neural network-based models, which typically represent SDEs through a combination of deterministic and stochastic terms. However, these models usually lack interpretability and have difficulty in generalizing beyond their training domain. Here, this paper introduces the Finite Expression Method (FEX), a symbolic learning approach designed to derive interpretable mathematical representations of the deterministic component of SDEs. For the stochastic component, we integrate FEX with advanced generative modeling techniques to provide a comprehensive representation of SDEs. The numerical experiments on linear, nonlinear, and multidimensional SDEs demonstrate that FEX generalizes well beyond the training domain and delivers more accurate long-term predictions compared to neural network-based methods. The symbolic expressions identified by FEX not only improve prediction accuracy but also offer valuable scientific insights into the underlying dynamics of the systems.
Understanding Dynamical Systems in High-Dimensional Parameter Spaces (Final Report)
In the last few decades, computational models, especially of the complex dynamical systems of greatest interest, i.e., climate, high energy density science, combustion, etc. have become increasingly accurate and sophisticated. Consequently, they are now routinely used to predict outcomes of experiments or to inform design decisions. However, many such simulations depend on a (large) number of parameters in order to, for example, approximate physical phenom whose exact solution is unknown or describe material properties in situations with little or no experimental data. Choosing these parameters and understanding how the outcomes of such a simulation changes with respect to the inputs are rapidly becoming some of the primary challenges in many application areas. Simulations have long been too complex for humans to fully comprehend and thus are often treated as black-box systems. The inputs to the systems are typically a collection of parameters with expert-selected ranges, and the outputs are various quantities of interest (QoIs) describing the performance of the system. Therefore, the fundamental challenge can be described as one of understanding a high-dimensional (many input parameters), multi-variate (many QoIs) function. Unfortunately, existing analysis tools are only of limited use for this task as most techniques can neither deal with many input dimensions nor offer opportunities to jointly study several QoIs. Furthermore, data analysis has long relied on an exploratory process in which scientist first visualize the available data, for example, using scatter plots or 3D renderings, form new hypotheses on some phenomena of interest, and subsequently test these hypotheses using dedicated analysis or statistics approaches. However, for high-dimensional functions this exploration is difficult if not impossible using existing tools as there are too many QoIs to simultaneously observe, and visualizations of high-dimensional spaces are abstract and difficult to understand if they exist at all. The goal of this project is to build new analysis techniques and tool that offer high level insights for exploring high-dimensional scientific data. And to extend existing approach beyond single scalar functions and instead consider multiple QoIs simultaneously.
Computing the non-Markovian coarse-grained interactions derived from the Mori–Zwanzig formalism in molecular systems: Application to polymer melts
Memory effects are often presented during coarse-graining of a complex dynamical system. In particular, a generalized Langevin equation (GLE) for the coarse-grained (CG) system arises in the context of Mori–Zwanzig formalism. Upon a pairwise decomposition, GLE can be reformulated into its pairwise version, i.e., non-Markovian dissipative particle dynamics (DPD). GLE models the dynamics of a single coarse particle, while DPD considers the dynamics of many interacting CG particles, with both CG systems governed by non-Markovian interactions. We compare two different methods for the practical implementation of the non-Markovian interactions in GLE and DPD systems. In particular, a direct evaluation of the non-Markovian (NM) terms is performed in LE-NM and DPD-NM models, which requires the storage of historical information that significantly increases computational complexity. Alternatively, we use a few auxiliary variables in LE-AUX and DPD-AUX models to replace the non-Markovian dynamics with a Markovian dynamics in a higher dimensional space, leading to a much reduced memory footprint and computational cost. In our numerical benchmarks, the GLE and non-Markovian DPD models are constructed from molecular dynamics (MD) simulations of star-polymer melts. Results show that a Markovian dynamics with auxiliary variables successfully generates equivalent non-Markovian dynamics consistent with the reference MD system, while maintaining a tractable computational cost. Also, transient subdiffusion of the star-polymers observed in the MD system can be reproduced by the coarse-grained models. The non-interacting particle models, LE-NM/AUX, are computationally much cheaper than the interacting particle models, DPD-NM/AUX. Yet, the pairwise models with momentum conservation are more appropriate for correctly reproducing the long-time hydrodynamics characterised by an algebraic decay in the velocity autocorrelation function.
Development of a Reference Governor-based Control Scheme for Integrated Energy Systems
The concept of an Integrated Energy System (IES) is meant to combine different energy technologies in synergistic ways to achieve a more secure and economical energy supply. The RAVEN-based HYBRID framework is used to find the optimal installed capacity and the optimal economical dispatch of each component of the IES. In practice what is achievable is limited by not only the inherent physical response characteristics of individual plant units but also by equipment operating limits. The inherent response characteristics are presently accounted for in the RAVEN-based HYBRID framework by dynamic models for plant units. The equipment operating limits, which are not yet addressed in HYBRID, are the subject of this report. These limits are set to achieve a design service life for a plant unit and are based on material degradation in the presence of mechanical fatigue induced by thermal cycling and by structure creep resulting from elevated temperature conditions. In general, the plant unit designer determines and specifies the allowable operating range of key process variables. It is then up to the integrated system design engineer to ensure those conditions are met. This can be accomplished through a combination of limiting the demand rates made on the plant unit and designing a control system that limits variations of key process variables. The typical control system design procedure, however, is not well suited for achieving this for complex dynamic systems such as encountered in integrated energy systems where multiple plant units must be simultaneously coordinated. This report describes how equipment operating limitations in an IES can be formally met through a control system design procedure know as the Reference Governor. The approach augments the existing controllers in a plant unit with an advance controller that modifies the issued setpoints so that selected process variables remain within limits. The design procedure is applied to the reference multi-unit system that has been the focus of Modeling and Simulation task in the Integrated Energy System program. Results describing application to the problem of limiting key process variables during operational transients for two of the plant units, the balance of plant and the gas turbine, are then presented. It is demonstrated that the Reference Governor method is an effective means for ensuring equipment operating constraints can be met in an IES. The software that implements the method has been made accessible to others on the Modeling and Simulation upload site.
Machine-learned closure of URANS for stably stratified turbulence: connecting physical timescales & data hyperparameters of deep time-series models
Stably stratified turbulence (SST), a model that is representative of the turbulence found in the oceans and atmosphere, is strongly affected by fine balances between forces and becomes more anisotropic in time for decaying scenarios. Moreover, there is a limited understanding of the physical phenomena described by some of the terms in the Unsteady Reynolds-Averaged Navier–Stokes (URANS) equations—used to numerically simulate approximate solutions for such turbulent flows. Rather than attempting to model each term in URANS separately, it is attractive to explore the capability of machine learning (ML) to model groups of terms, i.e. to directly model the force balances. We develop deep time-series ML for closure modeling of the URANS equations applied to SST. We consider decaying SST which are homogeneous and stably stratified by a uniform density gradient, enabling dimensionality reduction. We consider two time-series ML models: long short-term memory and neural ordinary differential equation. Both models perform accurately and are numerically stable in a posteriori (online) tests. Furthermore, we explore the data requirements of the time-series ML models by extracting physically relevant timescales of the complex system. We find that the ratio of the timescales of the minimum information required by the ML models to accurately capture the dynamics of the SST corresponds to the Reynolds number of the flow. The current framework provides the backbone to explore the capability of such models to capture the dynamics of high-dimensional complex dynamical system like SST flows.
Estimating the state of a geophysical system with sparse observations: time delay methods to achieve accurate initial states for prediction
The problem of forecasting the behavior of a complex dynamical system through analysis of observational time-series data becomes difficult when the system expresses chaotic behavior and the measurements are sparse, in both space and/or time. Despite the fact that this situation is quite typical across many fields, including numerical weather prediction, the issue of whether the available observations are "sufficient" for generating successful forecasts is still not well understood. An analysis by Whartenby et al. (2013) found that in the context of the nonlinear shallow water equations on a β plane, standard nudging techniques require observing approximately 70 % of the full set of state variables. Here we examine the same system using a method introduced by Rey et al. (2014a), which generalizes standard nudging methods to utilize time delayed measurements. Here, we show that in certain circumstances, it provides a sizable reduction in the number of observations required to construct accurate estimates and high-quality predictions. In particular, we find that this estimate of 70 % can be reduced to about 33 % using time delays, and even further if Lagrangian drifter locations are also used as measurements.
A systematic comparison of machine learning methods for modeling of dynamic processes applied to combustion emission rate modeling
Ten established, data-driven dynamic algorithms are surveyed and a practical guide for understanding these methods generated. Existing Python programming packages for implementing each algorithm are acknowledged, and the model equations necessary for prediction are presented. A case study on a coal-fired power plant’s NO x emission rates is performed, directly comparing each modeling method’s performance on a mutual system. Each model is evaluated by its root mean squared error (RMSE) on out-of-sample future horizon predictions. Optimal hyperparameters are identified using either an exhaustive search or genetic algorithm. The top five model structures of each method are used to recursively predict future NO x emission rates over a 60-step time horizon. The RMSE at each future timestep is determined, and the recursive output prediction trends compared against measurements in time. The GRU neural network is identified as the best candidate for representing the system, demonstrating accurate and stable predictions across the future horizon by all considered models, while satisfactory performance was observed in several of the ARX/NARX formulations. Finally, these efforts have contributed 1) a concise resource of multiple proven dynamic machine learning methods, 2) a practical guide explaining the use of these methods, effectively lowering the “barrier-to-entry” of deploying such models in control systems, 3) a comparison study evaluating each method’s performance on a mutual system, 4) demonstration of accurate multi-timestep emissions modeling suitable for systems-level control, and 5) generalizable results demonstrating the suitability of each method for prediction over a multi-step future horizon to other complex dynamic systems.
Data-driven prediction of battery cycle life before capacity degradation
Accurately predicting the lifetime of complex, nonlinear systems such as lithium-ion batteries is critical for accelerating technology development. However, diverse aging mechanisms, significant device variability and dynamic operating conditions have remained major challenges. We generate a comprehensive dataset consisting of 124 commercial lithium iron phosphate/graphite cells cycled under fast-charging conditions, with widely varying cycle lives ranging from 150 to 2,300 cycles. Using discharge voltage curves from early cycles yet to exhibit capacity degradation, we apply machine-learning tools to both predict and classify cells by cycle life. Our best models achieve 9.1% test error for quantitatively predicting cycle life using the first 100 cycles (exhibiting a median increase of 0.2% from initial capacity) and 4.9% test error using the first 5 cycles for classifying cycle life into two groups. This work highlights the promise of combining deliberate data generation with data-driven modelling to predict the behaviour of complex dynamical systems.
Scientific and Technical Basis of Application of Nanostructured Materials Science and Nanoelectronics in Systems of Electromechanical Energy Converters for Special Purposes
We consider the current state of a completely new area of science; microsystem electromechanics. Тhe wide spectra of its practical applications and prospects for further development are analyzed. We discuss in detail two main directions of creating micro- and nanoelectromechanical converters as basic elements of microsystem electromechanics: “top down” and “down top.” We also describe the main technological methods in constructing basic functional elements of microsystem electromechanics, areas of their application in traditional and new technologies (information and computer technologies, medicine, aerospace, rocket and artillery systems, etc.). At the same time, we discuss some key issues of generalized physical and mathematical modeling of microelectromechanical and nanoelectromechanical systems (MEMS and NEMS). A new generalized approach is proposed to study dynamic and energy characteristics of MEMS and NEMS as complex dynamic systems with binary conjugate subsystems. Based on proposed theoretical principles and models, the possibilities of studying electrophysical characteristics of biological nanostructures have been considered.
Variational multiscale reinforcement learning for discovering reduced order closure models of nonlinear spatiotemporal transport systems
Abstract A central challenge in the computational modeling and simulation of a multitude of science applications is to achieve robust and accurate closures for their coarse-grained representations due to underlying highly nonlinear multiscale interactions. These closure models are common in many nonlinear spatiotemporal systems to account for losses due to reduced order representations, including many transport phenomena in fluids. Previous data-driven closure modeling efforts have mostly focused on supervised learning approaches using high fidelity simulation data. On the other hand, reinforcement learning (RL) is a powerful yet relatively uncharted method in spatiotemporally extended systems. In this study, we put forth a modular dynamic closure modeling and discovery framework to stabilize the Galerkin projection based reduced order models that may arise in many nonlinear spatiotemporal dynamical systems with quadratic nonlinearity. However, a key element in creating a robust RL agent is to introduce a feasible reward function, which can be constituted of any difference metrics between the RL model and high fidelity simulation data. First, we introduce a multi-modal RL to discover mode-dependant closure policies that utilize the high fidelity data in rewarding our RL agent. We then formulate a variational multiscale RL (VMRL) approach to discover closure models without requiring access to the high fidelity data in designing the reward function. Specifically, our chief innovation is to leverage variational multiscale formalism to quantify the difference between modal interactions in Galerkin systems. Our results in simulating the viscous Burgers equation indicate that the proposed VMRL method leads to robust and accurate closure parameterizations, and it may potentially be used to discover scale-aware closure models for complex dynamical systems.
A fast reduced model for a shell-and-tube based latent heat thermal energy storage heat exchanger and its application for cost optimal design by nonlinear programming
Numerical simulation of latent heat thermal energy storage (LHTES) systems plays a fundamental role in studying the physical process and guiding the engineering design. Discretization of the PDEs describing the nonlinear solidification/melting process of phase change materials (PCMs) leads to a large-scale complex dynamics system, where the system behavior depends on a set of parameters. In a design setting, repeated model evaluations are required over the set of parameters results in significant computational burden. In this paper, an explicit analytic solution was built for the propagation of the solidification front in a cylindrical coordinate. The analytic solution approach is further employed to develop a low computational reduced model (RM) as a module for a shell-and-tube based LHTES heat exchanger. The levelized Cost of Energy (LCOE) is used as a design metric and the RM model is used to apply system-level constraints in the nonlinear programming formulation that facilitates efficient global optimal design of the PCM properties, flow conditions and tube geometries. The use of LCOE as the design metric prevents over design of the heat transfer rate and also establishes a fair ground for evaluation of different thermal storage technologies and their integrated applications with other systems. Optimal results showed that a higher effectiveness results in a higher LCOE; the velocity of the HTF and the length of the channel are highly correlated with each other; both larger PCM latent energy and conductivity result in lower LCOE.
On fast simulation of dynamical system with neural vector enhanced numerical solver
The large-scale simulation of dynamical systems is critical in numerous scientific and engineering disciplines. However, traditional numerical solvers are limited by the choice of step sizes when estimating integration, resulting in a trade-off between accuracy and computational efficiency. To address this challenge, we introduce a deep learning-based corrector called Neural Vector (NeurVec), which can compensate for integration errors and enable larger time step sizes in simulations. Our extensive experiments on a variety of complex dynamical system benchmarks demonstrate that NeurVec exhibits remarkable generalization capability on a continuous phase space, even when trained using limited and discrete data. NeurVec significantly accelerates traditional solvers, achieving speeds tens to hundreds of times faster while maintaining high levels of accuracy and stability. Moreover, NeurVec’s simple-yet-effective design, combined with its ease of implementation, has the potential to establish a new paradigm for fast-solving differential equations based on deep learning.
On Building Predictive Digital Twin Incorporating Wave Predicting Capabilities: Case Study on UMaine Experimental Campaign - FOCAL
The response of floating wind turbines (FWT) are susceptible to stochastic wave variations. For the optimal operation of FWT, a comprehensive understanding of the phaseresolved wave dynamics and the consequential system response is crucial for real-time monitoring and control. A multi-variate, multi-step, long short term memory (MLSTM), a type of recurrent neural network (RNN) is used to capture complex system dynamics for real-time application. Results indicate that the integration of a wave prediction-reconstruction (WRP) model substantially enhances prediction accuracy by 50% on average relative to the baseline model. The improvement is consistent across various wave extremity and prediction horizons, thereby significantly broadening the scope for timely and precise predictive capabilities.
Dynamic Control of Sodium Cold Trap Purification Temperature Using LSTM System Identification
This study investigates the dynamic regulation of the sodium cold trap purification temperature at Argonne National Laboratory’s liquid sodium test facility, employing long short-term memory (LSTM) system identification techniques. The investigation introduces an innovative hybrid approach by integrating model predictive control (MPC) based on first principles dynamic models with a multi-step time–frequency LSTM model in predicting the temperature profiles of a sodium cold trap purification system. The long short-term memory–model predictive controller (LSTM-MPC) model employs a sliding window scheme to gather training samples for multi-step prediction, leveraging historical data to construct predictive models that capture the non-linearities of the complex system dynamics without explicitly modeling the underlying physical processes. The performance of the LSTM-MPC and MPC were evaluated through simulation experiments, where both models were assessed on their capacity to maintain the cold trap temperature within predefined set-points while minimizing deviations and overshoots. Results obtained show how the data-driven LSTM-MPC model demonstrates stability and adaptability. In contrast, the traditional MPC model exhibits irregularities, particularly evident as overshoots around set-point limits, which can potentially compromise its effectiveness over long prediction time intervals. The findings obtained offer valuable insights into integrating data-driven techniques for enhancing real-time monitoring systems.
X-ray diffraction from nonuniformly stretched helical molecules
The fibrous proteins in living cells are exposed to mechanical forces interacting with other subcellular structures. X-ray fiber diffraction is often used to assess deformation and movement of these proteins, but the analysis has been limited to the theory for fibrous molecular systems that exhibit helical symmetry. However, this approach cannot adequately interpret X-ray data from fibrous protein assemblies where the local strain varies along the fiber length owing to interactions of its molecular constituents with their binding partners. To resolve this problem a theoretical formulism has been developed for predicting the diffraction from individual helical molecular structures nonuniformly strained along their lengths. This represents a critical first step towards modeling complex dynamical systems consisting of multiple helical structures using spatially explicit, multi-scale Monte Carlo simulations where predictions are compared with experimental data in a `forward' process to iteratively generate ever more realistic models. Here the effects of nonuniform strains and the helix length on the resulting magnitude and phase of diffraction patterns are quantitatively assessed. Examples of the predicted diffraction patterns of nonuniformly deformed double-stranded DNA and actin filaments in contracting muscle are presented to demonstrate the feasibly of this theoretical approach.
Quantum coherence–driven self-organized criticality and nonequilibrium light localization
Self-organized criticality emerges in dynamical complex systems driven out of equilibrium and characterizes a wide range of classical phenomena in physics, geology, and biology. We report on a quantum coherence–controlled self-organized critical transition observed in the light localization behavior of a coherence-driven nanophotonic configuration. Our system is composed of a gain-enhanced plasmonic heterostructure controlled by a coherent drive, in which photons close to the stopped-light regime interact in the presence of the active nonlinearities, eventually synchronizing their dynamics. In this system, on the basis of analytical and corroborating full-wave Maxwell-Bloch computations, we observe quantum coherence–controlled self-organized criticality in the emergence of light localization arising from the synchronization of the photons. It is associated with two first-order phase transitions: one pertaining to the synchronization of the dynamics of the photons and the second pertaining to an inversionless lasing transition by the coherent drive. The so-attained light localization, which is robust to dissipation, fluctuations, and many-body interactions, exhibits scale-invariant power laws and absence of finely tuned control parameters. We also found that, in this nonequilibrium dynamical system, the effective critical “temperature” of the system drops to zero, whereupon one enters the quantum self-organized critical regime.
Technical note: Using long short-term memory models to fill data gaps in hydrological monitoring networks
Abstract. Quantifying the spatiotemporal dynamics in subsurface hydrological flows over a long time window usually employs a network of monitoring wells. However, such observations are often spatially sparse with potential temporal gaps due to poor quality or instrument failure. In this study, we explore the ability of recurrent neural networks to fill gaps in a spatially distributed time-series dataset. We use a well network that monitors the dynamic and heterogeneous hydrologic exchanges between the Columbia River and its adjacent groundwater aquifer at the U.S. Department of Energy's Hanford site. This 10-year-long dataset contains hourly temperature, specific conductance, and groundwater table elevation measurements from 42 wells with gaps of various lengths. We employ a long short-term memory (LSTM) model to capture the temporal variations in the observed system behaviors needed for gap filling. The performance of the LSTM-based gap-filling method was evaluated against a traditional autoregressive integrated moving average (ARIMA) method in terms of error statistics and accuracy in capturing the temporal patterns of river corridor wells with various dynamics signatures. Our study demonstrates that the ARIMA models yield better average error statistics, although they tend to have larger errors during time windows with abrupt changes or high-frequency (daily and subdaily) variations. The LSTM-based models excel in capturing both high-frequency and low-frequency (monthly and seasonal) dynamics. However, the inclusion of high-frequency fluctuations may also lead to overly dynamic predictions in time windows that lack such fluctuations. The LSTM can take advantage of the spatial information from neighboring wells to improve the gap-filling accuracy, especially for long gaps in system states that vary at subdaily scales. While LSTM models require substantial training data and have limited extrapolation power beyond the conditions represented in the training data, they afford great flexibility to account for the spatial correlations, temporal correlations, and nonlinearity in data without a priori assumptions. Thus, LSTMs provide effective alternatives to fill in data gaps in spatially distributed time-series observations characterized by multiple dominant frequencies of variability, which are essential for advancing our understanding of dynamic complex systems.