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NASA NTRS2024

Nodal Modeling of Liquid Propellant Feed and Pressurization System

Nodal or network flow modeling plays an important role in the design and operation of the feed and pressurization system of a liquid rocket engine. Model development and execution time is relatively short for nodal codes in comparison to the Navier-Stokes based CFD codes. Nodal models also allow the inclusion of several components into one model to predict the behavior of a larger system. Unlike CFD models, the nodal models do not need very fine discretization of the flow field because they use empirical correlations to model fluid friction and heat transfer. This paper presents several applications of nodal modeling of liquid propellant feed and pressurization systems using the Generalized Fluid System Simulation Program (GFSSP), a nodal code developed at NASA/ Marshall Space Flight Center. GFSSP discretizes the flow field into nodes which are connected by branches. The mass and energy conservation equations and the equation of state are solved to calculate pressure, temperature, and resident mass at the nodes. The momentum equations are solved at the branches to calculate flow rates. Applications include a) tank pressurization by inert gas as well as autogenous pressurization by gaseous propellant, b) submerged pressurization by helium, c) self-pressurization due to boil-off of cryogenic propellant, d) chilldown of a transfer line of a cryogenic propellant feed system, and e) chilldown and filling of a cryogenic tank. Each of the above-mentioned models was verified and validated by comparing with test data.

Nodal Model↗

NASA NTRS2024

Nodal Modeling of Liquid Propellant Feed and Pressurization System

Nodal Model↗

NASA NTRS1994

A simple hyperbolic model for communication in parallel processing environments

We introduce a model for communication costs in parallel processing environments called the 'hyperbolic model,' which generalizes two-parameter dedicated-link models in an analytically simple way. Dedicated interprocessor links parameterized by a latency and a transfer rate that are independent of load are assumed by many existing communication models; such models are unrealistic for workstation networks. The communication system is modeled as a directed communication graph in which terminal nodes represent the application processes that initiate the sending and receiving of the information and in which internal nodes, called communication blocks (CBs), reflect the layered structure of the underlying communication architecture. The direction of graph edges specifies the flow of the information carried through messages. Each CB is characterized by a two-parameter hyperbolic function of the message size that represents the service time needed for processing the message. The parameters are evaluated in the limits of very large and very small messages. Rules are given for reducing a communication graph consisting of many to an equivalent two-parameter form, while maintaining an approximation for the service time that is exact in both large and small limits. The model is validated on a dedicated Ethernet network of workstations by experiments with communication subprograms arising in scientific applications, for which a tight fit of the model predictions with actual measurements of the communication and synchronization time between end processes is demonstrated. The model is then used to evaluate the performance of two simple parallel scientific applications from partial differential equations: domain decomposition and time-parallel multigrid. In an appropriate limit, we also show the compatibility of the hyperbolic model with the recently proposed LogP model.

Stoica, Ion↗

DOE OSTI2022

Resilient Operating Constraints for Power Distribution Systems under Setpoint Attacks

Integration and operation of distributed generation (DG) and energy storage (ES) in power distribution systems are enabled by communication networks and embedded sensor and control devices that increase the vulnerability of the systems to cyber-threats, broadening the attack surface and making adversary actions more unpredictable. This paper proposes a methodology that uses ellipsoidal approximations to quantify the potential damage caused by successful attacks that affect, directly or indirectly, the desired operation setpoints and may drive the power distribution operation to unsafe states by violating the limits of voltage or line flows. More specifically, a new methodology is introduced to find the optimal non-symmetric operating constraints that can be imposed to each DG and ES in order to guarantee that the power distribution system is resilient to any malicious setpoints. The proposed method takes as inputs the system topology, DG and ES capabilities, and load limits to solve a convex optimization problem formulated using linear matrix inequalities (LMIs) and the power flow equations. The proposed solution is agnostic to the attacker's action or load profile and it does not require any assumption about the location or means of the attack. The numerical results on a test distribution feeder with several DG and ES illustrate how the proposed resilient operating constraints guarantee the security of the power distribution system under setpoint attacks.

Giraldo, Jairo↗

DOE OSTI2020

nPINNs: Nonlocal physics-informed neural networks for a parametrized nonlocal universal Laplacian operator. Algorithms and applications

Physics-informed neural networks (PINNs) are effective in solving inverse problems based on differential and integro-differential equations with sparse, noisy, unstructured, and multifidelity data. PINNs incorporate all available information, including governing equations (reflecting physical laws), initial-boundary conditions, and observations of quantities of interest, into a loss function to be minimized, thus recasting the original problem into an optimization problem. In this paper, we extend PINNs to parameter and function inference for integral equations such as nonlocal Poisson and nonlocal turbulence models, and we refer to them as nonlocal PINNs (nPINNs). The contribution of the paper is three-fold. First, we propose a unified nonlocal Laplace operator, which converges to the classical Laplacian as one of the operator parameters, the nonlocal interaction radius $\delta$ goes to zero, and to the fractional Laplacian as $\delta$ goes to infinity. This universal operator forms a super-set of classical Laplacian and fractional Laplacian operators and, thus, has the potential to fit a broad spectrum of data sets. We also provide theoretical convergence rates with respect to $\delta$ and verify them via numerical experiments. Second, we use nPINNs to estimate the two parameters, $\delta$ and $\alpha$, characterizing the kernel of the unified operator. The strong non-convexity of the loss function yielding multiple (good) local minima reveals the occurrence of the operator mimicking phenomenon, that is, different pairs of estimated parameters could produce multiple solutions of comparable accuracy. Third, we propose another nonlocal operator with spatially variable order $\alpha(y)$, which is more suitable for modeling turbulent Couette flow. Our results show that nPINNs can jointly infer this function as well as $\delta$. More importantly, these parameters exhibit a universal behavior with respect to the Reynolds number, a finding that contributes to our understanding of nonlocal interactions in wall-bounded turbulence.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

nPINNs: nonlocal Physics-Informed Neural Networks for a parametrized nonlocal universal Laplacian operator. Algorithms and Applications

Physics-informed neural networks (PINNs) are effective in solving inverse problems based on differential and integro-differential equations with sparse, noisy, unstructured, and multifidelity data. PINNs incorporate all available information, including governing equations (reflecting physical laws), initial-boundary conditions, and observations of quantities of interest, into a loss function to be minimized, thus recasting the original problem into an optimization problem. In this paper, we extend PINNs to parameter and function inference for integral equations such as nonlocal Poisson and nonlocal turbulence models, and we refer to them as nonlocal PINNs (nPINNs). The contribution of the paper is three-fold. First, we propose a unified nonlocal Laplace operator, which converges to the classical Laplacian as one of the operator parameters, the nonlocal interaction radius δ goes to zero, and to the fractional Laplacian as δ goes to infinity. This universal operator forms a super-set of classical Laplacian and fractional Laplacian operators and, thus, has the potential to fit a broad spectrum of data sets. We provide theoretical convergence rates with respect to δ and verify them via numerical experiments. Second, we use nPINNs to estimate the two parameters, δ and α, characterizing the kernel of the unified operator. The strong non-convexity of the loss function yielding multiple (good) local minima reveals the occurrence of the operator mimicking phenomenon, that is, different pairs of estimated parameters could produce multiple solutions of comparable accuracy. Third, we propose another nonlocal operator with spatially variable order α(γ), which is more suitable for modeling turbulent Couette flow. Our results show that nPINNs can jointly infer this function as well as δ. More importantly, these parameters exhibit a universal behavior with respect to the Reynolds number, a finding that contributes to our understanding of nonlocal interactions in wall-bounded turbulence.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2021

A multi-dimensional parametric study of variability in multi-phase flow dynamics during geologic CO 2 sequestration accelerated with machine learning

Successful geologic CO 2 storage projects depend on numerical simulations to predict reservoir performance during site selection, injection verification, and post-injection monitoring phases of the project. These numerical simulations solve non-linear sets of coupled partial differential equations, while accounting for multi-phase fluid dynamics on the basis of constitutive equations that are embedded into the solution scheme. As a consequence, individual simulations often require tens to hundreds of hours to complete on high-performance computing clusters. Moreover, laboratory experiments reveal that parametric functions for capillary pressure and relative permeability exhibit substantial variability, even within the same rock type. This combination of computational expense and wide-ranging parametric variability means that there remains substantial uncertainty in the behavior of multi-phase CO 2 -water systems, particularly in the context of feedbacks between relative permeability and capillary pressure. To bridge this knowledge gap, here we develop a novel workflow that utilizes physics-based numerical simulation to train an artificial neural network (ANN) emulator for interrogating the multivariate parameter space that governs both capillary pressure and relative permeability. With this approach, the ANN is trained to emulate both fluid pressure distribution and CO 2 saturation, which are then interrogated quantitatively to generate parametric response surface mappings with high-fidelity resolution. Results from this study initially show that capillary entry pressure is the dominant control on both CO 2 plume geometry and fluid pressure propagation when considering the combined effects of capillary pressure and relative permeability, particularly when phase interference is low and residual CO 2 saturation is high. Moreover, the ANN emulator provides tremendous computational speed-up by computing 2691 individual simulations in several minutes; whereas, the same simulation ensemble would have required ~3 years of simulation time using only physics-based simulation methods (25,000 times speed up).

58 GEOSCIENCES↗

NASA NTRSDate not supplied

Harmonized Sentinel-1 SAR Global River Geometry and Inundation Database

Satellite-based observations on river geometries are sporadic in time, space, or both. Most satellite-based surface water maps, river widths, water surface elevations (WSE), slopes, and bathymetry are asynchronized in time and space. The current configuration of satellites such as Sentinel-6 measured the WSE but is missing the river width, slopes, and depths. To advance hydrological sciences research, there is a need to produce a harmonized time series of river geometry data of non-SWOT satellites in partnership with the upcoming SWOT mission. The SWOT satellite will measure river width, height, and slope but missing river depth measurements in space and time. Further, none of these current satellites measure the WSE, river width, and slopes synchronously. In this work, we use the Sentinel-1 SAR satellite data archive from 2015 to the present to create a global river width and surface water database at the reach scale. A modified version of the Sentinel SAR surface water classification algorithm from ASF is used to quantify the surface water extent on the stream approximately every six days (at the equator) at 10m spatial resolution globally. This 10m water mask is fed into a workflow to quantify the river widths, surface water inundations, slopes, and synthetic bathymetry in SWORD (SWOT River Database) stream networks. A Satellite HAND is used to address the cloud obscured surface water observations using a trained machine learning algorithm. We use WSE derived from the Global Water Monitor from NASA GSFC, Hydroweb from LEGOS, and ICESat-2 to harmonize the WSE observation. And Landsat-8/9 and Sentinel-2 water observations to fill the gaps in the Sentinel-1 SAR database. We use Congo River Basin as a test case where we have more than 500 radar altimetry-based WSE, continuous series of Sentinel-1, ICESat-2, Landsat-8/9, and Sentinel-2 observations. A Congo River hydrologic model is used to generate the streamflow discharge. The satellite observed river reaches are assimilated with the stream flows computed by the routing models. And the downstream reaches in the river network without satellite observations get optimized for discharge/river geometry at each observation cycle. Our final product is a harmonized river geometry dataset (reach's water extent, WSE, slope, synthetic bathymetry) for Congo Basin's SWORD reaches.

Chandana Gangodagamage↗

DOE OSTI2021

Machine learning accelerated turbulence modeling of transient flashing jets

We report modeling the sudden depressurization of superheated liquids through nozzles is a challenge because the pressure drop causes rapid flash boiling of the liquid. The resulting jet usually demonstrates a wide range of structures, including ligaments and droplets, due to both mechanical and thermodynamic effects. As the simulation comprises increasingly numerous phenomena, the computational cost begins to increase. One way to moderate the additional cost is to use machine learning surrogacy for specific elements of the calculation. This study presents a machine learning-assisted computational fluid dynamics approach for simulating the atomization of flashing liquids accounting for distinct stages, from primary atomization to secondary breakup to small droplets using the Σ - Y model coupled with the homogeneous relaxation model. Notably, the models for thermodynamic non-equilibrium (HRM) and Σ - Y are coupled, for the first time, with a deep neural network that simulates the turbulence quantities, which are then used in the prediction of superheated liquid jet atomization. The data-driven component of this method is used for turbulence modeling, avoiding the solution of the two-equation turbulence model typically used for Reynolds-averaged Navier-Stokes simulations for these problems. Both the accuracy and speed of the hybrid approach are evaluated, demonstrating adequate accuracy and at least 25% faster computational fluid dynamics simulations than the traditional approach. This acceleration suggests that perhaps additional components of the calculation could be replaced for even further benefit. Published under an exclusive license by AIP Publishing.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2020

Steady-State Analysis of Microgrid Distributed Control Under Denial of Service Attacks

Distributed control is an effective method to coordinate multiple components in a microgrid. It usually has mandatory requirements for communication graph topology; changing the topology unexpectedly could disrupt the operation of a distributed controller. A denial of service (DoS) attack blocks communication channels to cause variations in communication graph. Unforeseen topology changes render a controller to receive incomplete system information. Control decisions made based on incorrect information can lead to significantly reduced grid supporting capability. As a result, the vital power balance between generation and load may be challenged. Determining power balance requires steady-state power flow analysis; however, the cyber-induced steady-state analysis is not well studied in the literature. Here, in this article, we show that a cyber–physical microgrid with unknown DoS attacks has augmented uncertain power flow equations, whose special structure creates new computational issue to determining power balance. To deal with the issue, we develop a novel cyber-induced microgrid model: the cyber–physical system is transformed into a pure physical model, where the disruption effects of DoS in the cyber layer are shown to be equivalent to the variations of the physical model parameters. With the transformed system, we develop a sufficient condition to ascertain the system power balance under unknown DoS attacks. In addition, practical insights for resilient distributed control design are shown as well.

24 POWER TRANSMISSION AND DISTRIBUTION↗

DOE OSTI2024

Generative Physics-Informed Neural Network Solving Multi-Scale and Multi-Phase Plasma Chemical Flow Field

Low-temperature plasmas (LTPs) are non-equilibrium systems with near-room-temperature gas and highly energetic electrons. This makes them ideal for delicate applications in biomedicine and semiconductor manufacturing, enabling processes like wound healing, sterilization, etching, and plasma-enhanced chemical vapor deposition without thermal damage. However, LTPs involve complex chemistries, with hundreds of species and thousands of reactions, complicating their diagnosis, prediction, and control. Conventional diagnostics, such as Fourier-transform infrared spectroscopy (FTIR), laser-induced fluorescence (LIF), and optical emission spectroscopy (OES), offer limited species detection, while mass spectrometry (MS) struggles with low-sensitivity species. Additionally, LTP simulations face multi-scale challenges, as macroscopic fluid dynamics and microscopic particle collisions operate on vastly different timescales. To address these issues, we developed an artificial intelligence (AI) based diagnostic system: a generative physics-informed neural network (PINN-Gen) that can predict spatially resolved species concentrations and temperatures in LTPs by integrating experimental data from planar LIF with microscopic plasma chemical kinetics and macroscopic fluid mechanics, including plasma-liquid interactions at the interface between two phases. PINN-Gen solves no equations but checks the errors of physical laws by substituting the output from neural network, and the comparison with the experimental results. Thus, it naturally avoids the multi-scale difficulty of numerical simulations and predicts the results of conventionally unsolvable multi-scale and multi-phase problems. The real-time prediction will be robust due to the physical information used in the training of such a neural network, and only very limited input of condition required due to its generative feature.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2024

Forecasting high-dimensional spatio-temporal systems from sparse measurements

This paper introduces a new neural network architecture designed to forecast high-dimensional spatio-temporal data using only sparse measurements. The architecture uses a two-stage end-to-end framework that combines neural ordinary differential equations (NODEs) with vision transformers. Initially, our approach models the underlying dynamics of complex systems within a low-dimensional space; and then it reconstructs the corresponding high-dimensional spatial fields. Many traditional methods involve decoding high-dimensional spatial fields before modeling the dynamics, while some other methods use an encoder to transition from high-dimensional observations to a latent space for dynamic modeling. In contrast, our approach directly uses sparse measurements to model the dynamics, bypassing the need for an encoder. This direct approach simplifies the modeling process, reduces computational complexity, and enhances the efficiency and scalability of the method for large datasets. We demonstrate the effectiveness of our framework through applications to various spatio-temporal systems, including fluid flows and global weather patterns. Although sparse measurements have limitations, our experiments reveal that they are sufficient to forecast system dynamics accurately over long time horizons. Our results also indicate that the performance of our proposed method remains robust across different sensor placement strategies, with further improvements as the number of sensors increases. This robustness underscores the flexibility of our architecture, particularly in real-world scenarios where sensor data is often sparse and unevenly distributed.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

Physics guided machine learning using simplified theories

Recent applications of machine learning, in particular deep learning, motivate the need to address the generalizability of the statistical inference approaches in physical sciences. In this Letter, we introduce a modular physics guided machine learning framework to improve the accuracy of such data-driven predictive engines. The chief idea in our approach is to augment the knowledge of the simplified theories with the underlying learning process. To emphasize their physical importance, our architecture consists of adding certain features at intermediate layers rather than in the input layer. To demonstrate our approach, we select a canonical airfoil aerodynamic problem with the enhancement of the potential flow theory. We include the features obtained by a panel method that can be computed efficiently for an unseen configuration in our training procedure. By addressing the generalizability concerns, our results suggest that the proposed feature enhancement approach can be effectively used in many scientific machine learning applications, especially for the systems where we can use a theoretical, empirical, or simplified model to guide the learning module.

42 ENGINEERING↗

NASA NTRS2007

Partitioning Evapotranspiration in Semiarid Grassland and Shrubland Ecosystems Using Diurnal Surface Temperature Variation

The encroachment of woody plants in grasslands across the Western U.S. will affect soil water availability by altering the contributions of evaporation (E) and transpiration (T) to total evapotranspiration (ET). To study this phenomenon, a network of flux stations is in place to measure ET in grass- and shrub-dominated ecosystems throughout the Western U.S. A method is described and tested here to partition the daily measurements of ET into E and T based on diurnal surface temperature variations of the soil and standard energy balance theory. The difference between the mid-afternoon and pre-dawn soil surface temperature, termed Apparent Thermal Inertia (I(sub A)), was used to identify days when E was negligible, and thus, ET=T. For other days, a three-step procedure based on energy balance equations was used to estimate Qe contributions of daily E and T to total daily ET. The method was tested at Walnut Gulch Experimental Watershed in southeast Arizona based on Bowen ratio estimates of ET and continuous measurements of surface temperature with an infrared thermometer (IRT) from 2004- 2005, and a second dataset of Bowen ratio, IRT and stem-flow gage measurements in 2003. Results showed that reasonable estimates of daily T were obtained for a multi-year period with ease of operation and minimal cost. With known season-long daily T, E and ET, it is possible to determine the soil water availability associated with grass- and shrub-dominated sites and better understand the hydrologic impact of regional woody plant encroachment.

Moran, M. Susan↗

NASA NTRS2005

Towards Rocket Engine Components with Increased Strength and Robust Operating Characteristics

High-energy rotating machines, powering liquid propellant rocket engines, are subject to various sources of high and low cycle fatigue generated by unsteady flow phenomena. Given the tremendous need for reliability in a sustainable space exploration program, a fundamental change in the design methodology for engine components is required for both launch and space based systems. A design optimization system based on neural-networks has been applied and demonstrated in the redesign of the Space Shuttle Main Engine (SSME) Low Pressure Oxidizer Turbo Pump (LPOTP) turbine nozzle. One objective of the redesign effort was to increase airfoil thickness and thus increase its strength while at the same time detuning the vane natural frequency modes from the vortex shedding frequency. The second objective was to reduce the vortex shedding amplitude. The third objective was to maintain this low shedding amplitude even in the presence of large manufacturing tolerances. All of these objectives were achieved without generating any detrimental effects on the downstream flow through the turbine, and without introducing any penalty in performance. The airfoil redesign and preliminary assessment was performed in the Exploration Technology Directorate at NASA ARC. Boeing/Rocketdyne and NASA MSFC independently performed final CFD assessments of the design. Four different CFD codes were used in this process. They include WIL DCA T/CORSAIR (NASA), FLUENT (commercial), TIDAL (Boeing Rocketdyne) and, a new family (AardvarWPhantom) of CFD analysis codes developed at NASA MSFC employing LOX fluid properties and a Generalized Equation Set formulation. Extensive aerodynamic performance analysis and stress analysis carried out at Boeing Rocketdyne and NASA MSFC indicate that the redesign objectives have been fully met. The paper presents the results of the assessment analysis and discusses the future potential of robust optimal design for rocket engine components.

Marcu, Bogdan↗

DOE OSTI2023

PANDEMIC: Occupancy driven predictive ventilation control to minimize energy consumption and infection risk

During the SARS-CoV-2 (COVID-19) pandemic, governments around the world have formulated policies requiring ventilation systems to operate at a higher outdoor fresh air flow rate for a sufficient time, which has led to a sharp increase in building energy consumption. Therefore, it is necessary to identify an energy-efficient ventilation strategy to reduce the risk of infection. In this study, we developed an occupant-number-based model predictive control (OBMPC) algorithm for building ventilation systems. First, we collected the occupancy and Heating, ventilation, and air conditioning system (HVAC) data from March to July 2021. Then, four different models (Auto regression moving average-based multilayer perceptron (ARMA_MLP), Recurrent neural networks (RNN), Long short-term memory networks (LSTM), and Nonhomogeneous Markov with change points detection (NH_Markov)) were used to predict the number of room occupants from 15 min to 24 h ahead with an interval output. We found that each model could predict the number of occupants with 85% accuracy using a one-person offset. Furthermore, the accuracy of 15 min of the ahead prediction could reach 95% with a one-person offset, but none of them could track abrupt changes. The occupancy prediction results were used to calculate the ventilation demand using the Wells-Riley equation, and the upper bound can maintain an infection risk lower than 2% for 93% of the day. This OBMPC model could reduce the coil load by 52.44% and shift the peak load by 3 h up to 5 kW compared with 24 × 7 h full outdoor air (OA) system when people wear masks in the space. The occupancy prediction uncertainty could cause a 9% to 26% difference in demand ventilation, a 0.3°C to 2.4°C difference in zone temperature, a 28.5% to 44.5% difference in outdoor airflow rate, and a 10.7% to 28.2% difference in coil load.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗

NASA NTRS2003

Megawatt Electromagnetic Plasma Propulsion

The NASA Glenn Research Center program in megawatt level electric propulsion is centered on electromagnetic acceleration of quasi-neutral plasmas. Specific concepts currently being examined are the Magnetoplasmadynamic (MPD) thruster and the Pulsed Inductive Thruster (PIT). In the case of the MPD thruster, a multifaceted approach of experiments, computational modeling, and systems-level models of self field MPD thrusters is underway. The MPD thruster experimental research consists of a 1-10 MWe, 2 ms pulse-forming-network, a vacuum chamber with two 32 diffusion pumps, and voltage, current, mass flow rate, and thrust stand diagnostics. Current focus is on obtaining repeatable thrust measurements of a Princeton Benchmark type self field thruster operating at 0.5-1 gls of argon. Operation with hydrogen is the ultimate goal to realize the increased efficiency anticipated using the lighter gas. Computational modeling is done using the MACH2 MHD code, which can include real gas effects for propellants of interest to MPD operation. The MACH2 code has been benchmarked against other MPD thruster data, and has been used to create a point design for a 3000 second specific impulse (Isp) MPD thruster. This design is awaiting testing in the experimental facility. For the PIT, a computational investigation using MACH2 has been initiated, with experiments awaiting further funding. Although the calculated results have been found to be sensitive to the initial ionization assumptions, recent results have agreed well with experimental data. Finally, a systems level self-field MPD thruster model has been developed that allows for a mission planner or system designer to input Isp and power level into the model equations and obtain values for efficiency, mass flow rate, and input current and voltage. This model emphasizes algebraic simplicity to allow its incorporation into larger trajectory or system optimization codes. The systems level approach will be extended to the pulsed inductive thruster and other electrodeless thrusters at a future date.

Gilland, James↗

DOE OSTI2024

Divide and conquer: Learning chaotic dynamical systems with multistep penalty neural ordinary differential equations

Forecasting high-dimensional dynamical systems is a fundamental challenge in various fields, such as geosciences and engineering. Neural Ordinary Differential Equations (NODEs), which combine the power of neural networks and numerical solvers, have emerged as a promising algorithm for forecasting complex nonlinear dynamical systems. However, classical techniques used for NODE training are ineffective for learning chaotic dynamical systems. In this work, we propose a novel NODE-training approach that allows for robust learning of chaotic dynamical systems. Here, our method addresses the challenges of non-convexity and exploding gradients associated with underlying chaotic dynamics. Training data trajectories from such systems are split into multiple, non-overlapping time windows. In addition to the deviation from the training data, the optimization loss term further penalizes the discontinuities of the predicted trajectory between the time windows. The window size is selected based on the fastest Lyapunov time scale of the system. Multi-step penalty(MP) method is first demonstrated on Lorenz equation, to illustrate how it improves the loss landscape and thereby accelerates the optimization convergence. MP method can optimize chaotic systems in a manner similar to least-squares shadowing with significantly lower computational costs. Our proposed algorithm, denoted the Multistep Penalty NODE, is applied to chaotic systems such as the Kuramoto-Sivashinsky equation, the two-dimensional Kolmogorov flow, and ERA5 reanalysis data for the atmosphere. It is observed that MP-NODE provide viable performance for such chaotic systems, not only for short-term trajectory predictions but also for invariant statistics that are hallmarks of the chaotic nature of these dynamics.

Chaotic dynamical systems↗