Related papers: Higher-Order Automatic Differentiation Using Symbo…

Most nonlinear partial differential equation (PDE) solvers require the Jacobian matrix associated to the differential operator. In PETSc, this is typically achieved by either an analytic derivation or numerical approximation method such as…

Algorithms for the symbolic computation of polynomial conservation laws, generalized symmetries, and recursion operators for systems of nonlinear differential-difference equations (DDEs) are presented. The algorithms can be used to test the…

Automatic Differentiation (AD) allows to determine exactly the Taylor series of any function truncated at any order. Here we propose to use AD techniques for Monte Carlo data analysis. We discuss how to estimate errors of a general function…

A class of high-order numerical algorithms for Riesz derivatives are established through constructing new generating functions. Such new high-order formulas can be regarded as the modification of the classical (or shifted) Lubich's…

Executing complex manipulation in cluttered environments requires satisfying coupled geometric and temporal constraints. Although Spatio-Temporal Logic (SpaTiaL) offers a principled specification framework, its use in gradient-based…

We develop compositional learning algorithms for coupled dynamical systems, with a particular focus on electrical networks. While deep learning has proven effective at modeling complex relationships from data, compositional couplings…

For large-scale discrete-time algebraic Riccati equations (DAREs) with high-rank nonlinear and constant terms, the stabilizing solutions are no longer numerically low-rank, resulting in the obstacle in the computation and storage. However,…

This paper proposes a novel non-iterative method to solve power system differential algebraic equations (DAEs) using the differential transformation, a mathematical tool that can obtain power series coefficients by transformation rules…

Differential-algebraic equations (DAEs) are widely used for modeling of dynamical systems. The difficulty in solving numerically a DAE is measured by its differentiation index. For highly accurate simulation of dynamical systems, it is…

This paper introduces a scalar numerical differentiator, represented as a system of nonlinear differential equations of any high order. We derive the explicit solution for this system and demonstrate that, with a suitable choice of…

Where dual-numbers forward-mode automatic differentiation (AD) pairs each scalar value with its tangent derivative, dual-numbers /reverse-mode/ AD attempts to achieve reverse AD using a similarly simple idea: by pairing each scalar value…

In this work we present a theoretical model for differentiable programming. We construct an algebraic language that encapsulates formal semantics of differentiable programs by way of Operational Calculus. The algebraic nature of Operational…

Differentially-algebraic (D-algebraic) functions are solutions of polynomial equations in the function, its derivatives, and the independent variables. We revisit closure properties of these functions by providing constructive proofs. We…

Derivatives of computer graphics, image processing, and deep learning algorithms have tremendous use in guiding parameter space searches, or solving inverse problems. As the algorithms become more sophisticated, we no longer only need to…

Data augmentation (DA) algorithms are widely used for Bayesian inference due to their simplicity. In massive data settings, however, DA algorithms are prohibitively slow because they pass through the full data in any iteration, imposing…

Many algorithms for control, optimization and estimation in robotics depend on derivatives of the underlying system dynamics, e.g. to compute linearizations, sensitivities or gradient directions. However, we show that when dealing with…

Neural networks, especially the recent proposed neural operator models, are increasingly being used to find the solution operator of differential equations. Compared to traditional numerical solvers, they are much faster and more efficient…

We present a simple yet highly generalizable method for explaining interacting parts within a neural network's reasoning process. First, we design an algorithm based on cross derivatives for computing statistical interaction effects between…

Accurate image segmentation remains challenging, particularly in generating sharp, confident boundaries. While modern architectures have advanced the field, many of them still rely on standard loss functions like Cross-Entropy and Dice,…

A study is conducted to evaluate four derivative estimation methods when solving a large sparse nonlinear programming problem that arises from the approximation of an optimal control problem using a direct collocation method. In particular,…