scholarly journals A trajectory-based loss function to learn missing terms in bifurcating dynamical systems

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Rahel Vortmeyer-Kley ◽  
Pascal Nieters ◽  
Gordon Pipa
Keyword(s):  
Machine Learning ◽  
Dynamical Systems ◽  
State Space ◽  
Loss Function ◽  
Dynamical Behavior ◽  
Period Doubling ◽  
Training Data ◽  
Dynamical System Theory ◽  
Recent Developments ◽  
Period Doubling Bifurcations

AbstractMissing terms in dynamical systems are a challenging problem for modeling. Recent developments in the combination of machine learning and dynamical system theory open possibilities for a solution. We show how physics-informed differential equations and machine learning—combined in the Universal Differential Equation (UDE) framework by Rackauckas et al.—can be modified to discover missing terms in systems that undergo sudden fundamental changes in their dynamical behavior called bifurcations. With this we enable the application of the UDE approach to a wider class of problems which are common in many real world applications. The choice of the loss function, which compares the training data trajectory in state space and the current estimated solution trajectory of the UDE to optimize the solution, plays a crucial role within this approach. The Mean Square Error as loss function contains the risk of a reconstruction which completely misses the dynamical behavior of the training data. By contrast, our suggested trajectory-based loss function which optimizes two largely independent components, the length and angle of state space vectors of the training data, performs reliable well in examples of systems from neuroscience, chemistry and biology showing Saddle-Node, Pitchfork, Hopf and Period-doubling bifurcations.

DYNAMICS AND MODULATED CHAOS FOR TWO COUPLED OSCILLATORS

2004 ◽  
Vol 14 (01) ◽  
pp. 337-346 ◽  
Author(s):  
QINSHENG BI
Keyword(s):  
Coupled Oscillators ◽  
Dynamical Behavior ◽  
Period Doubling ◽  
Period Doubling Bifurcation ◽  
The Other ◽  
Van Der Pol ◽  
Parametrically Excited ◽  
Van Der Pol Oscillators ◽  
Period Doubling Bifurcations

The dynamical behavior of two coupled parametrically excited van der Pol oscillators is investigated in this paper. A special road to chaos is explored in detail. Period-doubling bifurcation associated with one of the frequencies of the system may be observed, the other frequency of the coupled oscillators plays a role in the evolution. It is found that one of the frequencies of the system contributes to the cascade of period-doubling bifurcations associated with the other frequency, which leads to a generalized modulated chaos.

scholarly journals On Instability Leading to Chaos in Dynamical Systems

1993 ◽  
Vol 132 ◽  
pp. 47-60
Author(s):  
L.M. Saha
Keyword(s):  
Dynamical Systems ◽  
Strange Attractor ◽  
Period Doubling ◽  
Stability Conditions ◽  
Stability Criteria ◽  
Linear Parameter ◽  
Standard Map ◽  
Taylor Map ◽  
Range Of Values ◽  
Period Doubling Bifurcations

AbstractInstability of orbits in dynamical systems leading to chaos has been reviewed briefly. Stability criteria for some unimodal mapping which provide various periodic regimes during the period doubling bifurcations has been discussed in detail. Stability conditions are reviewed for standard map (or Chirikov-Taylor map), and results obtained for range of values of the non-linear parameter appearing in the map have been studied. Strange attractor has also been discussed.

On the data synergy effect of large-sample multi-physics catchment modeling with machine learning

2021 ◽  
Author(s):  
Chaopeng Shen ◽  
Farshid Rahmani ◽  
Kuai Fang ◽  
Zhi Wei ◽  
Wen-Ping Tsai
Keyword(s):  
Machine Learning ◽  
Training Data ◽  
Similar Data ◽  
Continental Scale ◽  
Catchment Modeling ◽  
Multiple Regimes ◽  
Scale Models ◽  
Recent Developments ◽  
Data Synergy ◽  
Inherent Advantage

<p>Watersheds in the world are often perceived as being unique from each other, requiring customized study for each basin. Models uniquely built for each watershed, in general, cannot be leveraged for other watersheds. It is also a customary practice in hydrology and related geoscientific disciplines to divide the whole domain into multiple regimes and study each region separately, in an approach sometimes called regionalization or stratification. However, in the era of big-data machine learning, models can learn across regions and identify commonalities and differences. In this presentation, we first show that machine learning can derive highly functional continental-scale models for streamflow, evapotranspiration, and water quality variables. Next, through two hydrologic examples (soil moisture and streamflow), we argue that unification can often significantly outperform stratification, and systematically examine an effect we call data synergy, where the results of the DL models improved when data were pooled together from characteristically different regions and variables. In fact, the performance of the DL models benefited from some diversity in training data even with similar data quantity. However, allowing heterogeneous training data makes eligible much larger training datasets, which is an inherent advantage of DL. We also share our recent developments in advancing hydrologic deep learning and machine learning driven parameterization.</p>

scholarly journals Applying machine learning to study fluid mechanics

2022 ◽  
Author(s):  
Steven L. Brunton
Keyword(s):  
Machine Learning ◽  
Fluid Mechanics ◽  
Loss Function ◽  
Optimization Algorithm ◽  
Training Data ◽  
Data Driven ◽  
Physical Knowledge ◽  
Short Overview

Abstract This paper provides a short overview of how to use machine learning to build data-driven models in fluid mechanics. The process of machine learning is broken down into five stages: (1) formulating a problem to model, (2) collecting and curating training data to inform the model, (3) choosing an architecture with which to represent the model, (4) designing a loss function to assess the performance of the model, and (5) selecting and implementing an optimization algorithm to train the model. At each stage, we discuss how prior physical knowledge may be embedding into the process, with specific examples from the field of fluid mechanics. Graphic abstract

scholarly journals Nonlinear Dynamics Analysis of the Semiactive Suspension System with Magneto-Rheological Damper

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
Hailong Zhang ◽  
Enrong Wang ◽  
Fuhong Min ◽  
Ning Zhang ◽  
Chunyi Su ◽  
...  
Keyword(s):  
Dynamical Behavior ◽  
Period Doubling ◽  
Suspension System ◽  
Nonlinear Stability Analysis ◽  
Chaotic Motions ◽  
The Road ◽  
Road Profile ◽  
Road Excitation ◽  
Magneto Rheological ◽  
Period Doubling Bifurcations

This paper examines dynamical behavior of a nonlinear oscillator which models a quarter-car forced by the road profile. The magneto-rheological (MR) suspension system has been established, by employing the modified Bouc-Wen force-velocity (F-v) model of magneto-rheological damper (MRD). The possibility of chaotic motions in MR suspension is discovered by employing the method of nonlinear stability analysis. With the bifurcation diagrams and corresponding Lyapunov exponent (LE) spectrum diagrams detected through numerical calculation, we can observe the complex dynamical behaviors and oscillating mechanism of alternating periodic oscillations, quasiperiodic oscillations, and chaotic oscillations with different profiles of road excitation, as well as the dynamical evolutions to chaos through period-doubling bifurcations, saddle-node bifurcations, and reverse period-doubling bifurcations.

scholarly journals Order and chaos in hydrodynamic BL Her models

2013 ◽  
Vol 9 (S301) ◽  
pp. 489-490
Author(s):  
Radosław Smolec ◽  
Paweł Moskalik
Keyword(s):  
Dynamical Systems ◽  
Logistic Map ◽  
Period Doubling ◽  
Chaotic Behaviour ◽  
Turbulent Convection ◽  
Pulsating Stars ◽  
Period Doubling Bifurcations

AbstractMany dynamical systems of different complexity, e.g. 1D logistic map, the Lorentz equations, or real phenomena, like turbulent convection, show chaotic behaviour. Despite huge differences, the dynamical scenarios for these systems are strikingly similar: chaotic bands are born through the series of period doubling bifurcations and merge through interior crises. Within chaotic bands periodic windows are born through the tangent bifurcations, preceded by the intermittent behaviour. We demonstrate such behaviour in models of pulsating stars.

Scalable Approach to High Coverages on Oxides via Iterative Training of a Machine-Learning Algorithm

2019 ◽  
Author(s):  
Andrew Medford ◽  
Shengchun Yang ◽  
Fuzhu Liu
Keyword(s):  
Machine Learning ◽  
Chemical Potential ◽  
Learning Algorithm ◽  
Absolute Error ◽  
Low Energy ◽  
Training Data ◽  
High Coverage ◽  
Metal Compounds ◽  
Adsorption Energies ◽  
The Stability

Understanding the interaction of multiple types of adsorbate molecules on solid surfaces is crucial to establishing the stability of catalysts under various chemical environments. Computational studies on the high coverage and mixed coverages of reaction intermediates are still challenging, especially for transition-metal compounds. In this work, we present a framework to predict differential adsorption energies and identify low-energy structures under high- and mixed-adsorbate coverages on oxide materials. The approach uses Gaussian process machine-learning models with quantified uncertainty in conjunction with an iterative training algorithm to actively identify the training set. The framework is demonstrated for the mixed adsorption of CH<sub>x</sub>, NH<sub>x</sub> and OH<sub>x</sub> species on the oxygen vacancy and pristine rutile TiO<sub>2</sub>(110) surface sites. The results indicate that the proposed algorithm is highly efficient at identifying the most valuable training data, and is able to predict differential adsorption energies with a mean absolute error of ~0.3 eV based on <25% of the total DFT data. The algorithm is also used to identify 76% of the low-energy structures based on <30% of the total DFT data, enabling construction of surface phase diagrams that account for high and mixed coverage as a function of the chemical potential of C, H, O, and N. Furthermore, the computational scaling indicates the algorithm scales nearly linearly (N<sup>1.12</sup>) as the number of adsorbates increases. This framework can be directly extended to metals, metal oxides, and other materials, providing a practical route toward the investigation of the behavior of catalysts under high-coverage conditions.

Optimization of Diabetes Training DATA using Machine Learning Algorithms

2018 ◽  
Vol 6 (2) ◽  
pp. 283-286
Author(s):  
M. Samba Siva Rao ◽  
◽  
M.Yaswanth . ◽  
K. Raghavendra Swamy ◽  
◽  
...  
Keyword(s):  
Machine Learning ◽  
Learning Algorithms ◽  
Machine Learning Algorithms ◽  
Training Data

Virtual Screening Meets Deep Learning

2018 ◽  
Vol 15 (1) ◽  
pp. 6-28 ◽  
Author(s):  
Javier Pérez-Sianes ◽  
Horacio Pérez-Sánchez ◽  
Fernando Díaz
Keyword(s):  
Artificial Intelligence ◽  
Machine Learning ◽  
Deep Learning ◽  
Virtual Screening ◽  
Great Increase ◽  
New Drugs ◽  
Learning Approach ◽  
Screening Strategies ◽  
Computer Aided ◽  
Recent Developments

Background: Automated compound testing is currently the de facto standard method for drug screening, but it has not brought the great increase in the number of new drugs that was expected. Computer- aided compounds search, known as Virtual Screening, has shown the benefits to this field as a complement or even alternative to the robotic drug discovery. There are different methods and approaches to address this problem and most of them are often included in one of the main screening strategies. Machine learning, however, has established itself as a virtual screening methodology in its own right and it may grow in popularity with the new trends on artificial intelligence. Objective: This paper will attempt to provide a comprehensive and structured review that collects the most important proposals made so far in this area of research. Particular attention is given to some recent developments carried out in the machine learning field: the deep learning approach, which is pointed out as a future key player in the virtual screening landscape.

Comparative Analysis of Machine Learning Techniques Using Predictive Modeling

2020 ◽  
Vol 13 ◽  
Author(s):  
Ritu Khandelwal ◽  
Hemlata Goyal ◽  
Rajveer Singh Shekhawat
Keyword(s):  
Machine Learning ◽  
Comparative Analysis ◽  
Data Science ◽  
Training Data ◽  
Machine Learning Techniques ◽  
Future Trends ◽  
Data Set ◽  
Learning Stage ◽  
Learning Techniques ◽  
Different Types

Introduction: Machine learning is an intelligent technology that works as a bridge between businesses and data science. With the involvement of data science, the business goal focuses on findings to get valuable insights on available data. The large part of Indian Cinema is Bollywood which is a multi-million dollar industry. This paper attempts to predict whether the upcoming Bollywood Movie would be Blockbuster, Superhit, Hit, Average or Flop. For this Machine Learning techniques (classification and prediction) will be applied. To make classifier or prediction model first step is the learning stage in which we need to give the training data set to train the model by applying some technique or algorithm and after that different rules are generated which helps to make a model and predict future trends in different types of organizations. Methods: All the techniques related to classification and Prediction such as Support Vector Machine(SVM), Random Forest, Decision Tree, Naïve Bayes, Logistic Regression, Adaboost, and KNN will be applied and try to find out efficient and effective results. All these functionalities can be applied with GUI Based workflows available with various categories such as data, Visualize, Model, and Evaluate. Result: To make classifier or prediction model first step is learning stage in which we need to give the training data set to train the model by applying some technique or algorithm and after that different rules are generated which helps to make a model and predict future trends in different types of organizations Conclusion: This paper focuses on Comparative Analysis that would be performed based on different parameters such as Accuracy, Confusion Matrix to identify the best possible model for predicting the movie Success. By using Advertisement Propaganda, they can plan for the best time to release the movie according to the predicted success rate to gain higher benefits. Discussion: Data Mining is the process of discovering different patterns from large data sets and from that various relationships are also discovered to solve various problems that come in business and helps to predict the forthcoming trends. This Prediction can help Production Houses for Advertisement Propaganda and also they can plan their costs and by assuring these factors they can make the movie more profitable.

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