Lower bounds for nonparametric estimation of ordinary differential equations

Scientific machine learning is a burgeoning discipline which blends scientific computing and machine learning.Traditionally, scientific computing focuses on large-scale mechanistic models, usually differential equations, that are derived from scientific laws that simplified and explained phenomena.On the other hand, machine learning focuses on developing non-mechanistic data-driven models which require minimal knowledge and prior assumptions.The two sides have their pros and cons: differential equation models are great at extrapolating, the terms are explainable, and they can be fit with small data and few parameters.Machine learning models on the other hand require "big data" and lots of parameters but are not biased by the scientists ability to correctly identify valid laws and assumptions.However, the recent trend has been to merge the two disciplines, allowing explainable models that are data-driven, require less data than traditional machine learning, and utilize the knowledge encapsulated in centuries of scientific literature.The promise is to fuse a priori domain knowledge which doesn't fit into a "dataset", allow this knowledge to specify a general structure that prevents overfitting, reduces the number of parameters, and promotes extrapolatability, while still utilizing machine learning techniques to learn specific unknown terms in the model.This has started to be used for outcomes like automated hypothesis generation and accelerated scientific simulation.The purpose of this blog post is to introduce the reader to the tools of scientific machine learning, identify how they come together, and showcase the existing open source tools which can help one get started.We will be focusing on differentiable programming frameworks in the major languages for scientific machine learning: C++, Fortran, Julia, MATLAB, Python, and R.We will be comparing two important aspects: efficiency and composability.Efficiency will be taken in the context of scientific machine learning: by now most tools are well-optimized for the giant neural networks found in traditional machine learning, but, as will be discussed here, that does not necessarily make them efficient when deployed inside of differential equation solvers or when mixed with probabilistic programming tools.Additionally, composability is a key aspect of scientific machine learning since our toolkit is not ML in isolation.Our goal is not to do machine learning as seen in a machine learning conference (classification, NLP, etc.), and it's not to do traditional machine learning as applied to scientific data.Instead, we are putting ML models and techniques into the heart of scientific simulation tools to accelerate and enhance them.Our neural networks need to fully integrate with tools that simulate satellites and robotics simulators.They need to integrate with the packages that we use in our scientific work for verifying numerical accuracy, tracking units, estimating uncertainty, and much more.We need our neural networks to play nicely with existing packages for delay

Two Simplified Methods for Galloping of Iced Transmission Lines

The Laplace transform provides a technique for the solution of ordinary and partial differential equations for initial-value problems in which the number of independent variables is reduced by one. Ordinary differential equations become algebraic equations and equations such as the one-dimensional wave or diffusion equation become ordinary differential equations. The difficulty associated with the method manifests itself in the inversion which is required after the algebraic or ordinary differential equations have been solved. If the equations have suitable analytic solutions then the inversion may be effected either directly from tables or by using the complex inversion formula. If, however, such solutions are not suitable or if numerical solutions are obtained then inversion can cause serious problems. A numerical procedure developed some 25 years ago provides the opportunity for a straightforward numerical inversion. The method works well for ordinary and partial differential equations. In the latter case the resulting boundary-value problem may be solved by either the finite difference method, the finite element method or the boundary element method. The major advantage is that the method does not suffer from possible stability problems that may occur with the usual finite difference procedures in the time variable. The method is ideally suited to implementation on a spreadsheet.

The geometrical optical concept of rays is generalized for fields generated by any system of coupled non-linear partial differential equations of arbitrary order like the vector wave equations for (non)linear media, the Maxwell equations, the equations of magneto-hydrodynamics, the system of continuity (transport) equations for an inhomogeneous semiconductor, etc. It is shown that the pertinent solutions of such a system of partial differential equations can be obtained exactly from a set of ordinary non-linear differential equations, the so-called characteristic or ray equations. These characteristic (ray) equations are connected with the key equation of the theory, the generalized Hamilton–Jacobi equation. The generalized Hamilton–Jacobi equation is a first-order non-linear partial differential equation, a special case of which is known to be of great importance for classical mechanics. This equivalence between partial and ordinary differential equations implies an enormous simplification for the numerical evaluation of the original problem, as solving partial differential equations is much more difficult than solving ordinary differential equations. From the theoretical point of view this equivalence between partial and ordinary differential equations is also interesting for various reasons. We mention, for example, that interesting properties like stability, soliton behaviour, decay, growth, etc of the solutions of the field equations of physics can be directly deduced from the relatively easily obtained behaviour of the solutions of the ordinary non-linear differential equations for the ‘rays’. As an example of the potential powers of this new method we analyse the non-linear Schrödinger equation and derive its soliton solutions.

Output feedback vibration control of a string driven by a nonlinear actuator

Ordinary differential equations on singular spaces, Z. Bartosiewicz stability in delayed neural networks, J. Belair a condition on multi-existence of periodic solutions for a differential delay equation, Y. Cao control of global economic growth - will the centre hold?, E.N. Chukwu asymptotic behaviour of the Titchmarsh-Weyl coefficient for a coupled second order system, S.L. Clark stability problems for systems of nuclear reactors, C. Cordoneanu comparison theorems for disconjugate linear differential equations, M. Gaudenzi oscillation results for higher order nonlinear neutral delay equations with periodic coefficients, J.R. Graef, M.K. Grammatikopoulos and P.W. Spikes an implicit differential equation related to epidemic models, K.P. Hadeler and R. Shonkwiler the relationship between stability under disturbances and uniform stability in a periodic integrodifferential equation, Y. Hamaya on the asymptotic stability of the equilibrium of the damped oscillator, L. Hatvani on higher order nonlinear differential-difference equations, U. An Der Heiden vector field approximations flow homogencity, H. Hermes hopf bifurcation for a differential-difference equation from climate modeling, G. Hetzer shock layer behaviour for vector boundary value problems, S.J. Kirschvink properties of solutions of nth order equations, W.A.J. Kosmala small solutions to BVP's at resonance with nonhomogeneous nonlinearity, L. Lefton vibrational control of time delay systems, B. Lehman bifurcation set and compound eyes in a perturbed cubic Hamiltonian system, J. Li and Z. Lu gevrey character of formal solutions of nonlinear differential equations, X. Liu finite-difference schemes having the correct linear stability properties for all finite step-sizes, R.E. Mickens evolution of surface functionals and differential equations, Y. Li and J.S. Muldowney some remarks on stability properties in functional differential equations with infinite delay, S. Murakami and T. Yoshizawa periodic orbits of the Froeschle's map, A. Olvera and C. Vargas rotated vector fields, global families of limit cycles and Hilbert's 16th problem, L.M. Perko green's matrices and disconjugacy of a vector difference equation, A. Peterson the poincare manifold for the general case of a planar flow, W. Rivera hopf bifurcation in a class of ODE systems related to climate modeling, P.G. Schmidt on second order two point boundary value problems at resonance, M. Hihnala and S. Seikkala separatrix connections of quadratic gradient vector fields, D.S. Shafer. Part contents.

Liapunov Functions and Stability in Control Theory. Lecture Notes in Control and Information Sciences 267

The Green’s function is widely used in solving boundary value problems for differential equations, to which many mathematical and physical problems are reduced. In particular, solutions of partial differential equations by the Fourier method are reduced to boundary value problems for ordinary differential equations. Using the Green's function for a homogeneous problem, one can calculate the solution of an inhomogeneous differential equation. Knowing the Green's function makes it possible to solve a whole class of problems of finding eigenvalues in quantum field theory. The developed method for constructing the Green’s function of boundary value problems for ordinary linear differential equations is described. An algorithm and program for calculating the Green's function of boundary value problems for differential equations of the second and third orders in an explicit analytical form are presented. Examples of computing the Green's function for specific boundary value problems are given. The fundamental system of solutions of ordinary differential equations with singular points needed to construct the Green's function is calculated in the form of generalized power series with the help of the developed programs in the Maple environment. An algorithm is developed for constructing the Green's function in the form of power series for second-order and third-order differential equations with given boundary conditions. Compiled work programs in the Maple environment for calculating the Green functions of arbitrary boundary value problems for differential equations of the second and third orders. Calculations of the Green's function for specific third-order boundary value problems using the developed program are presented. The obtained approximate Green’s function is compared with the known expressions of the exact Green’s function and very good agreement is found

At the heart of many problems in mathematics, physics, and engineering lies the ordinary differential equation or its numerical equivalent, the ordinary finite difference equation. Ordinary differential equations arise not only in countless direct applications, but also occur indirectly, as reductions of partial differential equations (by way of separation of variables or by transform techniques for example; cf. Chaps. 9, 11). Likewise, the probably less familiar difference equations are of inherent interest (in probability, statistics, economics, etc.) but also appear as recurrence relations in connection with differential equations or as numerical approximations to differential equations.

Random ordinary differential equations (RODEs) are ordinary differential equations (ODEs) which have a stochastic process in their vector field functions. They have been used in a wide range of applications such as biology, medicine and engineering and play an important role in the theory of random dynamical systems. RODEs can be investigated pathwise as deterministic ODEs, however, the classical numerical methods for ODEs do not attain original order of convergence because the stochastic process has at most Hölder continuous sample paths and the resulting vector is also at most Hölder continuous in time. Recently, Jenzen & Kloeden derived new class of numerical methods for RODEs using integral versions of implicit Taylor-like expansions and developed arbitrary higher order schemes for RODEs. Their idea can be applied to random ordinary delay differential equations (RODDEs) by implementing Taylor-like expansions in the corresponding delay term. In this paper, numerical methods for RODDEs are systematically constructed based on Taylor-like expansions and they are applied to virus dynamics model with random fluctuations and time delay.

Rational design of organelle compartments in cells

Non-linear vibration and instability of multi-phase composite plate subjected to non-uniform in-plane parametric excitation: Semi-analytical investigation

Step forward on nonlinear differential equations with the Atangana–Baleanu derivative: Inequalities, existence, uniqueness and method

Chapter 1 - Introduction to Differential Equations

In this paper, Dzhumabaev's parameterization method is used to study ordinary differential equations with nonlocal boundary conditions. The parameter was introduced and the replacement was performed. The problem under consideration is divided into two parts: the first is the Cauchy problem for an ordinary differential equation, the second is a linear equation with respect to the introduced parameter. To determine the solution to the Cauchy problem, the Mikkusinsky operator method is used, based on the convolution theory. The Mikkusinsky operator method is an effective analytical tool used to solve ordinary differential and integral equations. This method is based on the theory of convolutions and allows finding solutions to equations using operator calculations. The main feature of Mikkusinsky's method is that it is considered algebraically, by transforming the differential equation using specially defined operators. Based on this method, using the values ​​of , we determine the solution to the Cauchy problem and, by placing it on the boundary conditions, we obtain a system of linear equations associated with the introduced parameters . Theorems on the solvability of the original nonlocal boundary value problem are formulated based on the solvability of the resulting equation with respect to its parameters. The results of the study demonstrate effective methods for solving differential equations with nonlocal conditions and clarify their theoretical foundations.