In the context of science, the well-known adage "a picture is worth a thousand words" might well be "a model is worth a thousand datasets." Scientific models, such as Newtonian physics or biological gene regulatory networks, are human-driven simplifications of complex phenomena that serve as surrogates for the countless experiments that validated the models. Recently, machine learning has been able to overcome the inaccuracies of approximate modeling by directly learning the entire set of nonlinear interactions from data. However, without any predetermined structure from the scientific basis behind the problem, machine learning approaches are flexible but data-expensive, requiring large databases of homogeneous labeled training data. A central challenge is reconciling data that is at odds with simplified models without requiring "big data". 
In this work we develop a new methodology, universal differential equations (UDEs), which augments scientific models with machine-learnable structures for scientifically-based learning. We show how UDEs can be utilized to discover previously unknown governing equations, accurately extrapolate beyond the original data, and accelerate model simulation, all in a time and data-efficient manner. This advance is coupled with open-source software that allows for training UDEs which incorporate physical constraints, delayed interactions, implicitly-defined events, and intrinsic stochasticity in the model. Our examples show how a diverse set of computationally-difficult modeling issues across scientific disciplines, from automatically discovering biological mechanisms to accelerating the training of physics-informed neural networks and large-eddy simulations, can all be transformed into UDE training problems that are efficiently solved by a single software methodology.

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Universal Differential Equations for Scientific Machine Learning

Analytical dynamics. L. A. Pars

L= distance between mass centers of satellites m1 and m2, tether length if m1 � m2 ,m L 0 = initial tether length, m M = total system mass, kg mi = mass of satellite i ,k g mt = tether mass, kg ri = position vector of mass mi from the system center of mass, m ˆ r1 = r1/a β, η = relative in-plane and out-of-plane swing angles between m1 and m2, respectively, deg βmax ,η max = maximum values of β and η, respectively, deg ˙ β0 = ˙ β at θ = 0 � Ha1 = maximum altitude decrease of m1 at apogee, m � Hi = altitude gain of satellite mass mi ,m � Hp2 = maximum altitude gain of m2 at perigee, m � L = decrease in the initial tether length L 0 during tether retrieval, m δ = ratio of in-plane swing rate ˙ β to the orbital rate � μ = gravitational constant, m 3 s −2 � = orbital rate, (μ/a 3 ) 1/2 , rad/s

Review on Dynamics and Control of Nonelectrodynamic Tethered Satellite Systems

We are concerned with families of dynamical systems which are related to generalized radix representations. The properties of these dynamical systems lead to new results on the characterization of bases of Pisot number systems as well as canonical number systems.

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Generalized radix representations and dynamical systems I

After presenting some basic introductory ideas concerning dimension reduction and reduced order modelling, an overview of the contents of the papers collected in this Special Issue of Nonlinear Dynamics is given.

Dimension Reduction of Dynamical Systems: Methods, Models, Applications

Tethered satellite systems (TSS) pose quite challenging problems concerning their modelling, derivation of the equations of motion, numerical simulation of their dynamics, deciding on stability of relative equilibria provided the system moves on a circular orbit around the Earth and the occurrence of chaotic dynamics. Moreover, for the processes of deployment or retrieval of one satellite from or to another satellite certain control strategies, for example time or energy optimal control, are necessary. All these problems are considered in this paper.

Modelling, Dynamics and Control of Tethered Satellite Systems

It is shown both by experiment and also by numerical simulation that for a vertically hanging folded chain the free part, if released, is falling faster than a free falling body under gravitational acceleration. A qualitative explanation of thisparadoxical phenomenon is given by showing that a downpulling force at the fold is created. In the simulation this force is also calculated quantitatively.

On the paradox of the free falling folded chain

A certain finiteness property of pisot number systems

The Use of the Adjoint Method for Solving Typical Optimization Problems in Multibody Dynamics

We construct a natural extension for each of Nakada's $\alpha$-continued fractions and show the continuity as a function of $\alpha$ of both the entropy and the measure of the natural extension domain with respect to the density function $(1+xy)^{-2}$. In particular, we show that, for all $0 < \alpha \le 1$, the product of the entropy with the measure of the domain equals $\pi^2/6$. As a key step, we give the explicit relationship between the $\alpha$-expansion of $\alpha-1$ and of $\alpha$.

Wolfgang Steiner

Papers

Modelling, Dynamics and Control of Tethered Satellite Systems

On the paradox of the free falling folded chain

A certain finiteness property of pisot number systems

The Use of the Adjoint Method for Solving Typical Optimization Problems in Multibody Dynamics

Natural extensions and entropy of $\alpha$-continued fractions