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All sources NASA NTRS DOE OSTI

DOE OSTI2021

Machine Learning Assisted Safety Modeling and Analysis of Advanced Reactors

With the advances in computational power and numerical methods, analysts can now rely on first-principle simulations to predict ultra-fine details in a variety of applications. Advances in machine learning (ML) have produced algorithms that can now learn high-level abstractions via hierarchical models. This project aims to leverage advances in ML techniques and the available high-resolution simulation data to develop a novel modeling and simulation (M\&S) methodology for reactor safety analysis. While application-agnostic ML techniques are available, complex physics constraints need to be incorporated into ML techniques to build ML-based closures for computationally efficient predictive simulations. This project intends to develop a physics-guided data-driven multi-scale methodology for M\&S of advanced reactors. The project focuses on thermal fluid (T/F) phenomena, which play major roles in advanced reactor safety. Specifically, we propose a data-driven coarse-mesh turbulence model based on local flow features for the transient analysis of thermal mixing and stratification in a sodium-cooled fast reactor (SFR). The model has a coarse-mesh setup to ensure computational efficiency, while it is trained by fine-mesh computational fluid dynamics (CFD) data with Reynolds-averaged Navier-Stokes (RANS) turbulence model to ensure accuracy. Three different neural networks are developed and tested for loss-of-flow transients in the hot pool of SFR, i.e. the densely connected convolutional neural network (DCNN), long-short-term-memory network based on proper orthogonal decomposition (POD-LSTM), and the DCNN informed by LSTM (DCNN-LSTM). The performances of these three neural networks are evaluated based on baseline models. The DCNN-LSTM model has been chosen for further hyperparameter optimization. Furthermore, based on a simplified two-dimensional case, uncertainty quantification (UQ) of the developed ML-based closure are investigated with three methods, i.e. Monte Carlo dropout, deep ensemble, and Bayesian neural network. The developed ML-based turbulent viscosity closure relation based on deep ensemble is then integrated into the system analysis module SAM and serves as a term in the conservation equations. Such a SAM-ML based procedure guarantees that the obtained results are consistent with the physical constraints of the thermal-fluid system. The SAM-ML simulation on the same loss-of-flow transient showed comparable accuracy with the CFD simulation but with a much coarser mesh setup. Last but not least, the ML-based closure improvement with the support of higher-fidelity data from large eddy simulation (LES) is discussed. As a first step towards this direction, a baseline LES simulation is performed to obtain comparable data with RANS results. Based on the early results, future investigation on further improving the ML-based closure is discussed. We believe the developed approach that combines scientific machine learning with nuclear system analysis code can benefit the advanced reactor community as more accurate safety analyses will better characterize reactor safety margins and reduce licensing efforts.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

DOE OSTI2026

Diesel Fuel Consumption in Prominent U.S. Open-Pit Mines: Site-Level Estimates

This report presents a comprehensive framework for estimating diesel fuel consumption and prices at open-pit mines in the United States. The framework includes transparent methods for calculating site-level diesel energy use when direct reporting is unavailable, and a structured confidence evaluation for each method. The framework is demonstrated to estimate current diesel consumption at 21 open-pit mines in the United States. Initial findings support ongoing efforts to strengthen the competitiveness and security of the U.S. industrial base by supporting data-driven supply chain analysis and decision-making, improved transparency in mining sector energy use, and targeted deployment of energy innovation and cost-reduction strategies. Future updates to the dataset—coupled with expanded data transparency and method validation—will help ensure that the findings remain relevant as the sector continues to evolve.

02 PETROLEUM↗

DOE OSTI2023

gLaSDI: Parametric physics-informed greedy latent space dynamics identification

A parametric adaptive physics-informed greedy Latent Space Dynamics Identification (gLaSDI) method is proposed for accurate, efficient, and robust data-driven reduced-order modeling of high-dimensional nonlinear dynamical systems. In the proposed gLaSDI framework, an autoencoder discovers intrinsic nonlinear latent representations of high-dimensional data, while dynamics identification (DI) models capture local latent-space dynamics. Here, an interactive training algorithm is adopted for the autoencoder and local DI models, which enables identification of simple latent-space dynamics and enhances accuracy and efficiency of data-driven reduced-order modeling. To maximize and accelerate the exploration of the parameter space for the optimal model performance, an adaptive greedy sampling algorithm integrated with a physics-informed residual-based error indicator and random-subset evaluation is introduced to search for the optimal training samples on the fly. Further, to exploit local latent-space dynamics captured by the local DI models for an improved modeling accuracy with a minimum number of local DI models in the parameter space, a -nearest neighbor convex interpolation scheme is employed. The effectiveness of the proposed framework is demonstrated by modeling various nonlinear dynamical problems, including Burgers equations, nonlinear heat conduction, and radial advection. The proposed adaptive greedy sampling outperforms the conventional predefined uniform sampling in terms of accuracy. Compared with the high-fidelity models, gLaSDI achieves 17 to 2,658× speed-up with 1 to 5% relative errors.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

On learning what to learn: Heterogeneous observations of dynamics and establishing possibly causal relations among them

Abstract Before we attempt to (approximately) learn a function between two sets of observables of a physical process, we must first decide what the inputs and outputs of the desired function are going to be. Here we demonstrate two distinct, data-driven ways of first deciding “the right quantities” to relate through such a function, and then proceeding to learn it. This is accomplished by first processing simultaneous heterogeneous data streams (ensembles of time series) from observations of a physical system: records of multiple observation processes of the system. We determine (i) what subsets of observables are common between the observation processes (and therefore observable from each other, relatable through a function); and (ii) what information is unrelated to these common observables, therefore particular to each observation process, and not contributing to the desired function. Any data-driven technique can subsequently be used to learn the input–output relation—from k-nearest neighbors and Geometric Harmonics to Gaussian Processes and Neural Networks. Two particular “twists” of the approach are discussed. The first has to do with the identifiability of particular quantities of interest from the measurements. We now construct mappings from a single set of observations from one process to entire level sets of measurements of the second process, consistent with this single set. The second attempts to relate our framework to a form of causality: if one of the observation processes measures “now,” while the second observation process measures “in the future,” the function to be learned among what is common across observation processes constitutes a dynamical model for the system evolution.

Sroczynski, David W.↗

DOE OSTI2021

Machine Learning for Automated Weld Quality Monitoring and Control (CRADA 461)

The emergence of Artificial Intelligence (AI) and Machine Learning (ML) methods may provide an alternative pathway to elicit the desired process-structure-property relationship. To achieve such relationship at an accelerated pace the use of data driven machine learning techniques are employed. This CRADA investigates the use and effectiveness of applying AI/ML techniques to a large dataset with the intention of creating a data-driven model that will predict processing parameters required to join lightweight materials at optimal performance.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

Adaptive Fault Detection Based on Neural Networks and Multiple Sampling Points for Distribution Networks and Microgrids

Smart networks such as microgrid (MG) and active distribution networks (ADN) have been recently playing an important role in power system operation. The design and implementation of appropriate protection systems for such networks must be addressed, which imposes new technical challenges. This paper presents the implementation and validation aspects of an adaptive fault detection strategy based on neural networks (NNs) and multiple sampling points for ADN and MG. The solution is implemented on an edge device. Artificial NNs are used to derive a data-driven model that uses only local measurements to detect fault states of the network without the need for communication infrastructure. Multiple sampling points are used to derive a data-driven model, which allows the generalization considering the implementation in physical systems. The adaptive fault detector model is implemented on a Jetson Nano system, which is a single-board computer (SBC) with a small Graphic Processing Unit (GPU) intended to run machine learning loads at the edge. The proposed method is tested in a physical, real-life, low-voltage network located at Universidad del Norte, Colombia. This testing network is based on the IEEE-13 Node Test Feeder scaled down to 220 V. The validation in a simulation environment shows the accuracy and dependability above 99.6%, while the real-time tests show the accuracy and dependability of 95.5% and 100%, respectively. Without hard-to-derive parameters, the easy-to-implement embedded model highlights the potential for real-life applications.

42 ENGINEERING↗

NASA NTRSDate not supplied

A Review of Options for a National Lab for Microgravity/Orbital Activities in LEO

NASA’s Office of Technology, Policy, and Strategy (OTPS) performs data-driven assessments to inform the Agency’s development of policy and strategy. OTPS works across NASA, the US government, industry, and academia to incorporate relevant stakeholder input to data-driven assessments regarding the future of NASA. OTPS conducted a study with the goal of identifying potential models to ensure that NASA and its government-sponsored user communities can successfully complete activities in low-Earth orbit (LEO) in a post-International Space Station (ISS) (i.e. 2030s) timeframe. Given the expectation that one or more Commercial LEO Destinations will replace the ISS by 2030, NASA leadership is considering next steps for microgravity research and other activities in LEO. This task is particularly challenging due to the uncertain future of the LEO ecosystem. Future scenarios may range from a robust environment with multiple platform options and new activities to an austere environment with few platform options and deflated momentum among user communities. Both extreme futures, along with any number of scenarios in between, have serious implications for the future of LEO and NASA’s role in it. It is therefore imperative that NASA consider its prioritized activities for the 2030 timeframe and identify potential business and operating models that will provide continuity for government-sponsored activities. The results of this NASA OTPS study will be presented, along with a discussion of how specific engagement models could enable NASA and other entities to achieve their goals in the 2030 timeframe.

Erica Marie Rodgers Heusinkveld↗

DOE OSTI2022

Physics-Informed Neural Network Solution of Point Kinetics Equations for a Nuclear Reactor Digital Twin

A digital twin (DT) for nuclear reactor monitoring can be implemented using either a differential equations-based physics model or a data-driven machine learning model. The challenge of a physics-model-based DT consists of achieving sufficient model fidelity to represent a complex experimental system, whereas the challenge of a data-driven DT consists of extensive training requirements and a potential lack of predictive ability. We investigate the performance of a hybrid approach, which is based on physics-informed neural networks (PINNs) that encode fundamental physical laws into the loss function of the neural network. We develop a PINN model to solve the point kinetic equations (PKEs), which are time-dependent, stiff, nonlinear, ordinary differential equations that constitute a nuclear reactor reduced-order model under the approximation of ignoring spatial dependence of the neutron flux. The PINN model solution of PKEs is developed to monitor the start-up transient of Purdue University Reactor Number One (PUR-1) using experimental parameters for the reactivity feedback schedule and the neutron source. The results demonstrate strong agreement between the PINN solution and finite difference numerical solution of PKEs. We investigate PINNs performance in both data interpolation and extrapolation. For the test cases considered, the extrapolation errors are comparable to those of interpolation predictions. Extrapolation accuracy decreases with increasing time interval.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

DOE OSTI2021

Scenario adaptive disruption prediction study for next generation burning-plasma tokamaks

Next generation High Performance (HP) tokamaks risk damage from unmitigated disruptions at high current and power. Achieving reliable disruption prediction for a device's HP operation based on its Low Performance (LP) data is a key to its success. In this letter, through explorative data analysis and dedicated numerical experiments on multiple existing tokamaks, we demonstrate how the operational regimes of tokamaks can affect the power of a trained disruption predictor. First, our results suggest data-driven disruption predictors trained on abundant LP discharges work poorly on the HP regime of the same tokamak, which is a consequence of the distinct distributions of the tightly correlated signals related to disruptions in these two regimes. Second, we find that matching operational parameters among tokamaks strongly improves cross-machine accuracy which implies our model learns from the underlying scalings of dimensionless physics parameters like q 95 , β p and confirms the importance of these parameters in disruption physics and cross machine domain matching from the data-driven perspective. Finally, our results show in the absence of HP data from the target devices, the best predictivity of the HP regime for the target machine can be achieved by combining LP data from the target with HP data from other machines. Furthermore, these results provide a possible disruption predictor development strategy for next generation tokamaks, such as ITER and SPARC, and highlight the importance of developing baseline scenario discharges of future tokamaks on existing machines to collect more relevant disruptive data.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2015

An efficient method for unfolding kinetic pressure driven VISAR data

Velocity Interferometer System for Any Reflector (VISAR) [Barker and Hollenbach, J. Appl. Phys. 43, 4669 (1972)] is a well-known diagnostic that is employed on many shock physics and pulsed-power experiments. With the VISAR diagnostic, the velocity on the surface of any metal flyer can be found. For most experiments employing VISAR, either a kinetic pressure [Grady, Mech. Mater. 29, 181 (1998)] or a magnetic pressure [Lemke et al., Intl J. Impact Eng. 38, 480 (2011)] drives the motion of the flyer. Moreover, reliable prediction of the time-dependent pressure is often a critical component to understanding the physics of these experiments. Although VISAR can provide a precise measurement of a flyer’s surface velocity, the real challenge of this diagnostic implementation is using this velocity to unfold the time-dependent pressure. The purpose of this study is to elucidate a new method for quickly and reliably unfolding VISAR data.

42 ENGINEERING↗

DOE OSTI2017

Machine Learning for Turbulence Modeling

This work was conducted as part of a Harry S. Truman Fellowship Laboratory Directed Research and Development project. The goal was to use machine learning methods to provide uncertainty quantification and model improvements for Reynolds Averaged Navier Stokes (RANS) turbulence models. For applications of interest in energy, safety, and security, it is critical to be able to model turbulence accurately. Current RANS models are unreliable for many flows of engineering relevance. Machine learning provides an avenue for developing improved models based on the data generated by high fidelity simulations. In this project, machine learning methods were used to predict when current RANS models would fail. They were also used to develop improved RANS closure models. A key aim was developing a tight feedback loop between scientific domain knowledge and data driven methods. To this end, a methodology for incorporating known invariance constraints into the machine learning models was proposed and evaluated. This work demonstrated that incorporating known constraints into the data driven models provided improved performance and reduced computational cost. This research represents one of the first applications of deep learning to turbulence modeling.

42 ENGINEERING↗

NASA NTRS1986

Petri net model for analysis of concurrently processed complex algorithms

This paper presents a Petri-net model suitable for analyzing the concurrent processing of computationally complex algorithms. The decomposed operations are to be processed in a multiple processor, data driven architecture. Of particular interest is the application of the model to both the description of the data/control flow of a particular algorithm, and to the general specification of the data driven architecture. A candidate architecture is also presented.

Stoughton, John W.↗

NASA NTRSDate not supplied

Anomaly Detection for the Roman Space Telescope Wide Field Instrument’s Science Data Processing Pipeline

The Roman Space Telescope (RST) Wide Field Instrument (WFI) will be utilizing a preliminary Science Data Processing (SDP) pipeline during its Integration and Test, and to some extent during Operations, to track basic statistics and identify known features such as cosmic rays, snowballs as well as possible anomalies in raw detector data. In our detectors, these anomalies appear as jumps in the ramp of a readout and are classified as cosmic rays if they appear as a streak or snowballs if they’re more circular. The WFI employs an array of 18 H4RG-10 detectors that collect image samples. Each set of raw frames within a non-destructive exposure is packaged by the SDP pipeline into image cubes for each detector. Each cube is a time series of 4096 × 4096 accumulating pixel frames. The preliminary analysis pipeline is used to locate anomalies in these time-series accumulation frames and identify the type of anomaly, either natural phenomena or detector characteristic. To compare different methods, we’ve implemented both heuristic-based and data-driven methods to identify anomalies. For the heuristic-based approach, we identify snowballs and cosmic rays by the size and shape of outlier pixel clusters between consecutive frames. For data driven methods, we evaluated a Convolutional Neural Network (CNN) model, and more traditional methods like Principal Component Analysis (PCA). CNN is a supervised learning/classification method. Thus, we used a labeled dataset of anomalies to perform segmentation of the image and identify anomalies. We used previously identified cosmic rays and snowballs to measure the accuracy and efficiency of the mentioned approaches. In evaluating these methods, we aim to pick the best fit for the SDP pipeline’s anomaly detection in terms of both performance and runtime.

Paul Horton↗

DOE OSTI2018

Voltron Compatible Whole Building Root-Fault Detection and Diagnosis

Buildings consume more than 40% of primary energy in the U.S. Malfunctioning sensors, components, and control systems, as well as degrading components in Heating, Ventilating and Air-conditioning (HVAC) and lighting systems are main reasons for energy waste and unsatisfactory indoor environment. Field studies have shown that an energy saving of 5-30% of total building energy consumption and improved indoor air quality can be achieved by simply applying automated fault detection and diagnosis (AFDD), followed by corrections even if these are not done in real time. Extensive research has been made on the development of component level AFDD tools in the past two decades. However, in recent years, the development of method for a whole building level AFDD has received much attention. Here, a whole building fault refers to a fault happening in one subsystem but impact on more than one subsystems, or has a significant impact on building performances, such as building’s energy consumption. Existing component level AFDD tools could fail to detect or give false alarm for faults that trigger abnormalities in multiple subsystems. Isolating such whole building faults, which have impacts on multiple subsystems, is also challenging by simply using component based AFDD solutions. On the other hand, it might be more cost-effective for medium- or small-sized commercial buildings that have subsystems (e.g. primary cooling subsystem, air loop subsystem, heating subsystem, lighting subsystem, etc.) to install one whole building level AFDD tool, rather than many component level AFDD tools, to focus on significant faults. The focus of this project is firstly to provide training opportunities in building AFDD research area for undergraduate students and to develop a workforce pipeline in building sciences. The objective of the proposed research activities is to develop cost-effective and scalable AFDD solutions for whole building level faults. The developed AFDD solution is VOLTTRONTM compatible to further increase its plug-n-play capacity and the market penetration. VOLTTRONTM is an open-source innovative distributed control and sensing software platform developed by the Pacific Northwest National Laboratory. A VOLTTRONTM compatible AFDD tool brings the benefits such as low-cost deployment, and better building to grid integration, etc. The performance of the developed whole building AFDD tools is evaluated using data collected from a mixed-use medium-sized university campus building. Another significant goal of this project is to engage undergraduate students in research activities and expose them to building control and AFDD fields. A demonstration building – Nesbitt Hall at Drexel University is identified for this study. This demonstration building has a typical HVAC system configuration, which includes a water-cooled chiller system, three variable air volume (VAV) air handling unit (AHU) systems, and a hydronic heating system. Faults, that are expected to have a whole building level impact, are artificially implemented in this demonstration building for three different seasons. Building data have been collected in this project, which include fault free building automation system (BAS) data and BAS data that contain system behaviors caused by artificially implemented and naturally occurred faults. These collected data are used to evaluate the developed whole building AFDD tools. A data-driven method, which includes a weather and schedule-based pattern matching (WPM) method and Feature based Principal Component Analysis (FPCA) method, is developed for whole building level fault detection. Parameter sensitivity tests, including snapshot window size, data sample searching pool size, etc., are implemented to evaluate the impacts of various WPM-FPCA parameters. A data-driven and expert knowledge/rule-based method using Bayesian Network (BN) and WPM is developed for whole building level fault diagnosis. The developed WPM-BN method, which includes a two-layer BN structure model and BN parameters, are either learned from baseline data or developed from expert knowledge. The developed whole building AFDD tools are evaluated using BAS data collected from the demonstration building. Using the evaluation data set, the WPM-FPCA fault detection method is evaluated under two different Principle Component (PC) retention rates, i.e., 0.65 retention rate and 0.95 retention rate. It is observed that when using 0.65 PC retention rate, the WPM-FPCA fault detection method achieves a 79% fault detection rate (this rate reaches to 85% when including the six make-up fault test cases discussed in the next bullet), while when using 0.95 PC retention rate, the WPM-FPCA fault detection method achieves 100% fault detection rate (this rate remains to be 100% when including the six make-up fault test cases). Under both PC retention rates, the WPM-FPCA fault detection method demonstrates 0% false alarm rates. Using the evaluation data set, the developed WPM-BN fault diagnosis method has successfully diagnosed thirteen (13) faults out of the fourteen (14) fault test cases. The one mis-diagnosed case exhibits fewer fault symptoms in different subsystems, when compared with other fault test cases. Overall speaking, the developed whole building AFDD solutions demonstrate satisfactory fault detection/diagnosis accuracy with very low false alarm rate. Lastly, a survey and market study has been performed to understand the existing AFDD market, including its gaps and needs.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗

DOE OSTI2022

Reliability estimation of an advanced nuclear fuel using coupled active learning, multifidelity modeling, and subset simulation

Tristructural isotropic (TRISO)-coated particle fuel is a robust nuclear fuel and determining its reliability is critical for the success of advanced nuclear technologies. However, TRISO failure probabilities are small and the associated computational models are expensive. Here we used coupled active learning, multifidelity modeling, and subset simulation to estimate the failure probabilities of TRISO fuels using several 1D and 2D models. With multifidelity modeling, we replaced expensive high-fidelity (HF) model evaluations with information fusion from two low-fidelity (LF) models. For the 1D TRISO models, we considered three multifidelity modeling strategies: only Kriging, Kriging LF prediction plus Kriging correction, and deep neural network (DNN) LF prediction plus Kriging correction. While the results across these multifidelity modeling strategies compared satisfactorily, strategies employing information fusion from two LF models called the HF model least often. Next, for the 2D TRISO model, we considered two multifidelity modeling strategies: DNN LF prediction plus Kriging correction (data-driven) and 1D TRISO LF prediction plus Kriging correction (physics-based). The physics-based strategy, as expected, consistently required the fewest calls to the HF model. However, the data-driven strategy had a lower overall simulation time since the DNN predictions are instantaneous, and the 1D TRISO model requires a non-negligible simulation time.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗

DOE OSTI2017

Machine learning in materials informatics: recent applications and prospects

Propelled partly by the Materials Genome Initiative, and partly by the algorithmic developments and the resounding successes of data-driven efforts in other domains, informatics strategies are beginning to take shape within materials science. These approaches lead to surrogate machine learning models that enable rapid predictions based purely on past data rather than by direct experimentation or by computations/simulations in which fundamental equations are explicitly solved. Data-centric informatics methods are becoming useful to determine material properties that are hard to measure or compute using traditional methods—due to the cost, time or effort involved—but for which reliable data either already exists or can be generated for at least a subset of the critical cases. Predictions are typically interpolative, involving fingerprinting a material numerically first, and then following a mapping (established via a learning algorithm) between the fingerprint and the property of interest. Fingerprints, also referred to as “descriptors”, may be of many types and scales, as dictated by the application domain and needs. Predictions may also be extrapolative—extending into new materials spaces—provided prediction uncertainties are properly taken into account. This article attempts to provide an overview of some of the recent successful data-driven “materials informatics” strategies undertaken in the last decade, with particular emphasis on the fingerprint or descriptor choices. The review also identifies some challenges the community is facing and those that should be overcome in the near future.

36 MATERIALS SCIENCE↗

DOE OSTI2022

A database of refractive indices and dielectric constants auto-generated using ChemDataExtractor

The ability to auto-generate databases of optical properties holds great potential for advancing optical research, especially with regards to the data-driven discovery of optical materials. An optical property database of refractive indices and dielectric constants is presented, which comprises a total of 49,076 refractive index and 60,804 dielectric constant data records on 11,054 unique chemicals. The database was auto-generated using the state-of-the-art natural language processing software, ChemDataExtractor, using a corpus of 388,461 scientific papers. The data repository offers a representative overview of the information on linear optical properties that resides in scientific papers from the past 30 years. Public availability of these data will enable a quick search for the optical property of certain materials. The large size of this repository will accelerate data-driven research on the design and prediction of optical materials and their properties. To the best of our knowledge, this is the first auto-generated database of optical properties from a large number of scientific papers. We provide a web interface to aid the use of our database.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2025

Acceleration of Power System Dynamic Simulations Using a Deep Equilibrium Layer and Neural ODE Surrogate

The dominant paradigm for power system dynamic simulation is to build system-level simulations by combining physics-based models of individual components. The sheer size of the system along with the rapid integration of inverter-based resources exacerbates the computational burden of running time domain simulations. Here, in this paper, we propose a data-driven surrogate model based on implicit machine learningspecifically deep equilibrium layers and neural ordinary differential equationsto learn a reduced order model of a portion of the full underlying system. The data-driven surrogate achieves similar accuracy and reduction in simulation time compared to a physics-based surrogate, without the constraint of requiring detailed knowledge of the underlying dynamic models. This work also establishes key requirements needed to integrate the surrogate into existing simulation workflows; the proposed surrogate is initialized to a steady state operating point that matches the power flow solution by design.

Neural ordinary differential equations↗