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The discovery of governing equations from scientific data has the potential to transform data-rich fields that lack well-characterized quantitative descriptions. Advances in sparse regression are currently enabling the tractable identification of both the structure and parameters of a nonlinear dynamical system from data. The resulting models have the fewest terms necessary to describe the dynamics, balancing model complexity with descriptive ability, and thus promoting interpretability and generalizability. This provides an algorithmic approach to Occam’s razor for model discovery. However, this approach fundamentally relies on an effective coordinate system in which the dynamics have a simple representation. In this work, we design a custom deep autoencoder network to discover a coordinate transformation into a reduced space where the dynamics may be sparsely represented. Thus, we simultaneously learn the governing equations and the associated coordinate system.We demonstrate this approach on several example high-dimensional systems with low-dimensional behavior. The resulting modeling framework combines the strengths of deep neural networks for flexible representation and sparse identification of nonlinear dynamics (SINDy) for parsimonious models. This method places the discovery of coordinates and models on an equal footing.

This paper presents a machine learning framework for Bayesian systems identification from noisy, sparse and irregular observations of nonlinear dynamical systems. The proposed method takes advantage of recent developments in differentiable programming to propagate gradient information through ordinary differential equation solvers and perform Bayesian inference with respect to unknown model parameters using Hamiltonian Monte Carlo sampling. This allows an efficient inference of the posterior distributions over plausible models with quantified uncertainty, while the use of sparsity-promoting priors enables the discovery of interpretable and parsimonious representations for the underlying latent dynamics. A series of numerical studies is presented to demonstrate the effectiveness of the proposed methods, including nonlinear oscillators, predator–prey systems and examples from systems biology. Taken together, our findings put forth a flexible and robust workflow for data-driven model discovery under uncertainty. All codes and data accompanying this article are available at https://bit.ly/34FOJMj.

Process mining with educational data has made use of various algorithms for model discovery, principally Alpha Miner, Heuristic Miner, and Evolutionary Tree Miner. In this study we propose the implementation of a new algorithm for educational data called Inductive Miner. We used data from the interactions of 101 university students in a course given over one semester on the Moodle 2.0 platform. Data was extracted from the platform's event logs; following preprocessing, the mining was carried out on 21,629 events to discover what models the various algorithms produced and to compare their fitness, precision, simplicity and generalization. The Inductive Miner algorithm produced the best results in the tests on this dataset, especially for fitness, which is the most important criterion in terms of model discovery. In addition, when we weighted the various metrics according to their importance, Inductive Miner continued to produce the best results. Inductive Miner is a new algorithm which, in addition to producing better results than other algorithms using our dataset, also provides valid models which can be interpreted in educational terms.

Real-time and model-predictive control promises to make urban drainage systems (UDS) adaptive, coordinated, and dynamically optimal. Though early implementations are promising, existing control algorithms have drawbacks in computational expense, trust, system-level coordination, and labor cost. Linear feedback control has distinct advantages in computational expense, interpretation, and coordination. However, current methods for building linear feedback controllers require calibrated software models. Here we present an automated method for generating tunable linear feedback controllers that require only system response data. The controller design consists of three main steps: (1) estimating the network connectivity using tools for causal inference, (2) identifying a linear, time-invariant (LTI) dynamical system which approximates the network, and (3) designing and tuning a feedback controller based on the LTI urban drainage system approximation. The flooding safety, erosion prevention, and water treatment performance of the method are evaluated across 190 design storms on a separated sewer model. Strong results suggest that the system knowledge required for generating effective, safe, and tunable controllers for UDS is surprisingly basic. This method allows near-turnkey synthesis of controllers solely from sensor data or reduction of process-based models.

Identifying governing equations in physical and biological systems from datasets remains a long-standing challenge across various scientific disciplines. Common methods like sparse identification of nonlinear dynamics (SINDy) often rely on precise derivative approximations, making them sensitive to data scarcity and noise. This study presents a novel data-driven framework by integrating high order implicit Runge-Kutta methods (IRKs) with the sparse identification, termed IRK-SINDy. The framework exhibits remarkable robustness to data scarcity and noise by relying on the A-stability of IRKs and consequently their fewer limitations on stepsize. Two methods for incorporating IRKs into sparse regression are introduced: one employs iterative schemes for numerically solving nonlinear algebraic system of equations, while the other utilizes deep neural networks to predict stage values of IRKs. The performance of IRK-SINDy is demonstrated through numerical experiments on synthetic data in benchmark problems with varied dynamical behaviors, including linear and nonlinear oscillators, the Lorenz system, and biologically relevant models like predator-prey dynamics, logistic growth, and the FitzHugh-Nagumo model. Results indicate that IRK-SINDy outperforms conventional SINDy and the RK4-SINDy framework, particularly under conditions of extreme data scarcity and noise, yielding interpretable and generalizable models.

Background Mathematics and Phy sics-based simulation models have the potential to help interpret and encapsulate biological phenomena in a computable and reproducible form. Similarly, comprehensive descriptions of such models help to ensure that such models are accessible, discoverable, and reusable. To this end, researchers have developed tools and standards to encode mathematical models of biological systems enabling reproducibility and reuse, tools and guidelines to facilitate semantic description of mathematical models, and repositories in which to archive, share, and discover models. Scientists can leverage these resources to investigate specific questions and hypotheses in a more efficient manner. Results We have comprehensively annotated a cohort of models with biological semantics. These annotated models are freely available in the Physiome Model Repository (PMR). To demonstrate the benefits of this approach, we have developed a web-based tool which enables users to discover models relevant to their work, with a particular focus on epithelial transport. Based on a semantic query, this tool will help users discover relevant models, suggesting similar or alternative models that the user may wish to explore or use. Conclusion The semantic annotation and the web tool we have developed is a new contribution enabling scientists to discover relevant models in the PMR as candidates for reuse in their own scientific endeavours. This approach demonstrates how semantic web technologies and methodologies can contribute to biomedical and clinical research. The source code and links to the web tool are available at https://github.com/dewancse/model-discovery-tool

The analysis of a timeseries can provide many new perspectives if it is accompanied by the assumption that the timeseries is generated from an underlying dynamical system. For example, statistical properties of the data can be related to measure theoretic aspects of the dynamics, and one can try to recreate the dynamics itself. The underlying dynamics could represent a natural phenomenon or a physical system, where the timeseries represents a sequence of measurements. In this paper, we present a completely data-driven framework to identify and model quasiperiodically driven dynamical systems (Q.P.D.) from the timeseries it generates. Q.P.D. are a special class of systems that are driven by a periodic source with multiple base frequencies. Such systems abound in nature, e.g., astronomy and traffic flow. Our framework reconstructs the dynamics into two components - the driving quasiperiodic source with generating frequencies; and the driven nonlinear dynamics. We make a combined use of a kernel-based harmonic analysis, kernel-based interpolation technique, and Koopman operator theory. Our framework provides accurate reconstructions and frequency identification for three real-world case studies.

This research is a quasi-experimental that aimed to compare the curiosity of students with discovery-based contextual models and direct learning models. The independent variables in this study are discovery-based contextual models and direct learning models, while the dependent variable is the students' curiosity in acid-base topic. The population in this study is all students of class XI MIA SMAN 1 Gowa, while the sample was class XI MIA 1 as experimental group I and class XI MIA 7 as experimental group II with the number of students each 30 people. The research data was obtained through an observation sheet and student activity sheet. The results of the descriptive analysis showed that the average value obtained by the experimental group I was 79 while in the experimental group II it was 68,63. The results of hypothesis testing using t-test values obtained at tcount = 5,99 and significance level α = 0,05 with df = 3 obtained ttable = 1,64. It shows that the curiosity of student that are taught by discovery-based contextual models was higher that their taugh by direct learning models in class XI MIA SMAN 1 Gowa on acid-base topic.

Despite recent achievements in the design and manufacture of advanced materials, the contributions from first-principles modeling and simulation have remained limited, especially in regards to characterizing how macroscopic properties depend on the heterogeneous microstructure. An improved ability to model and understand these multiscale and anisotropic effects will be critical in designing future materials, especially given rapid improvements in the enabling technologies of additive manufacturing and active metamaterials. In this review, we discuss recent progress in the data-driven modeling of dynamical systems using machine learning and sparse optimization to generate parsimonious macroscopic models that are generalizable and interpretable. Such improvements in model discovery will facilitate the design and characterization of advanced materials by improving efforts in (1) molecular dynamics, (2) obtaining macroscopic constitutive equations, and (3) optimization and control of metamaterials.

Oscillatory dynamics play a central role in the description of a broad spectrum of physical systems. While often well‐approximated by linear models, the essential long‐term evolution and stability of these systems are frequently determined by subtle nonlinearities. The characterization of such weakly nonlinear systems from observational data is a central challenge in system identification, a task made difficult by the immense disparity in magnitude between faint nonlinear signatures and the dominant linear response. Herein, we introduce Evolutionary Learning Oscillator with Weak Nonlinearity (EvLOWN), a data‐driven methodology for inferring the governing equations of weakly nonlinear oscillators from sparse and potentially noisy time‐series observations. We first demonstrate EvLOWN's superior accuracy and robustness on benchmark systems, then apply it to uncover the subtle on‐site and coupling potentials in fundamental models, including the Fermi‐Pasta‐Ulam and Klein‐Gordon chains, using only local measurements. Translating this framework to critical engineering applications, we reconstruct the orbital dynamics of the Tiangong and International Space Stations from public data, revealing nearly identical governing laws despite their distinct architectures. Furthermore, from wind‐tunnel experiments on a scaled suspension bridge, EvLOWN extracts the precise equations of motion capturing complex vortex‐induced vibrations. These results establish EvLOWN as a powerful tool for the data‐driven discovery of governing laws in complex systems where weak nonlinearities play a crucial yet subtle role.