Solving Ordinary Differential Equations

By S. Linge, Hans Petter Langtangen. Programming for Computations - Python, 2019

1,800 citations recorded by the source corpus.

Differential equations constitute one of the most powerful mathematical tools to understand and predict the behavior of dynamical systems in nature, engineering, and society. A dynamical system is some system with some state, usually expressed by a set of variables, that evolves in time. For example, an oscillating pendulum, the spreading of a disease, and the weather are examples of dynamical systems. We can use basic laws of physics, or plain intuition, to express mathematical rules that govern the evolution of the system in time. These rules take the form of differential equations. You are probably well experienced with equations, at least equations like \(ax+b=0\) or \(ax^{2}+bx+c=0\). Such equations are known as algebraic equations, and the unknown is a number. The unknown in a differential equation is a function, and a differential equation will almost always involve this function and one or more derivatives of the function. For example, \(f^{\prime}(x)=f(x)\) is a simple differential equation (asking if there is any function f such that it equals its derivative – you might remember that e x is a candidate).

Subfields: Artificial Intelligence, Information Systems, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/2b8561daa06ec6a44078fc22036aeb68cd8d173a

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Citations in the corpus

Solution formulas for differential Sylvester and Lyapunov equations (2018, methodology): “where αj and β are coefficients of the BDF method and can be found in [15].”
Frequency evaluation for exponentially fitted Runge-Kutta methods (2005, methodology): “It is interesting to check the algebraic order conditions well described in[13].”
Exact optimal values of step-size coefficients for boundedness of linear multistep methods (2017, methodology): “6 Actual proofs for the Adams–Bashforth methods The coefficients for the Adams–Bashforth methods are listed, for example, in [3].”
Neural Networks with Cheap Differential Operators (2019, methodology): “Efficient Jacobi Iterations in Implicit Linear Multistep Methods Ordinary differential equations (ODEs) parameterized by neural networks are typically solved using explicit methods such as Runge-Kutta 4(5) (Hairer and Peters, 1987).”
acados—a modular open-source framework for fast embedded optimal control (2019, background): “Within the family of single-step integration methods one typically distinguishes between explicit and implicit schemes [44].”
DifferentialEquations.jl – A Performant and Feature-Rich Ecosystem for Solving Differential Equations in Julia (2017, background): “These days, most processors are multi-core and many computers contain GPGPU [1] or Xeon Phi [17, 6] acceleration cards and thus taking advantage of the ever-present parallelism is key to achieving good performance.”
Enforcing Constraints for Time Series Prediction in Supervised, Unsupervised and Reinforcement Learning (2019, methodology): “For example, we can consider the simplest temporal discretization scheme, the forward Euler scheme [12], and discretize (1) with timestep ∆t to obtain”
Learning Differential Equations that are Easy to Solve (2020, methodology): “When h is sufficiently small, the numerical error of a mth-order method is bounded by ‖ẑ(t+ h)− z(t+ h)‖ ≤ ch for some constant c (Hairer et al., 1993).”
Consistent tangent operators for constitutive rate equations (2002, methodology): “For these three constants we have chosen the standard values from literature, see [16].”
Evolutionary generation of high‐order, explicit, two‐step methods for second‐order linear IVPs (2017, methodology): “The enumeration of order conditions follows from the theory of Nyström methods.(9,10,48) If we had to solve a scalar ODE of the type (Equation 2), then some of the elementary differentials appearing in Equation 3 may coincide.”
Improving and generalizing flow-based generative models with minibatch optimal transport (2023, neutral): “Specifically, in this case we let ¯ z := { z i := ( x i 0 , x i 1 ) } mi =1 , where z i are i.i.d. from q ( z ) and It takes roughly m times as long to compute the conditional target u t ( x | ¯ z ) but reduces the variance.”
From linear to nonlinear MPC: bridging the gap via the real-time iteration (2020, methodology): “This requires the numerical simulation of a nonlinear system of di ff erential equations (see Hairer, Nørsett, and Wanner (1993) and Hairer, Nørsett, and Wanner (1996) for a detailed overview).”
Reduced-order modeling of advection-dominated systems with recurrent neural networks and convolutional autoencoders (2020, methodology): “The reader is directed to [48] for greater discussion of the temporal integration scheme and [49] for details on WENO and the Riemann solver implementation in two-dimensional problems.”