Higher-Order Partial Differentiation

In this article, we shall extend the formalization of [10] to discuss higher-order partial differentiation of real valued functions. The linearity of this operator is also proved (refer to [10], [12] and [13] for partial differentiation).This work was supported by JSPS KAKENHI 22300285 and 23500029Endou Noboru - Nagano National College of Technology, JapanOkazaki Hiroyuki - Shinshu University, Nagano, JapanShidama Yasunari - Shinshu University, Nagano, JapanGrzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Czesław Bylinski. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.Czesław Bylinski. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Bylinski. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.Noboru Endou and Yasunari Shidama. Completeness of the real Euclidean space. Formalized Mathematics, 13(4):577-580, 2005.Noboru Endou, Yasunari Shidama, and Keiichi Miyajima. Partial differentiation on normed linear spaces Rn. Formalized Mathematics, 15(2):65-72, 2007, doi:10.2478/v10037-007-0008-5.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.Takao Inou´e, Noboru Endou, and Yasunari Shidama. Differentiation of vector-valued functions on n-dimensional real normed linear spaces. Formalized Mathematics, 18(4):207-212, 2010, doi: 10.2478/v10037-010-0025-7.Takao Inou´e, Adam Naumowicz, Noboru Endou, and Yasunari Shidama. Partial differentiation of vector-valued functions on n-dimensional real normed linear spaces. Formalized Mathematics, 19(1):1-9, 2011, doi: 10.2478/v10037-011-0001-x.Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.Keiichi Miyajima and Yasunari Shidama. Riemann integral of functions from R into Rn. Formalized Mathematics, 17(2):179-185, 2009, doi: 10.2478/v10037-009-0021-y.Keiko Narita, Artur Kornilowicz, and Yasunari Shidama. More on the continuity of real functions. Formalized Mathematics, 19(4):233-239, 2011, doi: 10.2478/v10037-011-0032-3.Takaya Nishiyama, Keiji Ohkubo, and Yasunari Shidama. The continuous functions on normed linear spaces. Formalized Mathematics, 12(3):269-275, 2004.Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Beata Perkowska. Functional sequence from a domain to a domain. Formalized Mathematics, 3(1):17-21, 1992.Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111-115, 1991.Yasunari Shidama. Banach space of bounded linear operators. Formalized Mathematics, 12(1):39-48, 2004.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990

Solving the Partial Differential Problems Using Maple

AbstractThis paper considers the partial differential problem of two types of multivariable functions and uses mathematical software Maple for verification. The infinite series forms of any order partial derivatives of these two types of multivariable functions can be obtained using binomial series and differentiation term by term theorem, which greatly reduce the difficulty of calculating their higher order partial derivative values. On the other hand, four examples are used to demonstrate the calculations

Automatic Evaluation of Higher-Order Partial Derivatives for Nonlocal Sensitivity Analysis

This proceedings paper surveys work on the FEED (Fast Efficient Evaluation of Derivatives) algorithm originally developed by Kalaba, Tesfatsion, and Wang (1983), with particular reference to the use of FEED for the implementation of nonlocal automated sensitivity techniques; see the articles below. The relationship of FEED to the automatic differentiation algorithms developed by other SIAM Workshop participants is clarified. Related work can be accessed here: http://www.econ.iastate.edu/tesfatsi/nasahome.htmAutomatic evaluation; higher-order partial derivatives; FEED; automated sensitivity analysis

Boundary-layer equations in generalized curvilinear coordinates

A set of higher-order boundary-layer equations is derived valid for three-dimensional compressible flows. The equations are written in a generalized curvilinear coordinate system, in which the surface coordinates are nonorthogonal; the third axis is restricted to be normal to the surface. Also, higher-order viscous terms which are retained depend on the surface curvature of the body. Thus, the equations are suitable for the calculation of the boundary layer about arbitrary vehicles. As a starting point, the Navier-Stokes equations are derived in a tensorian notation. Then by means of an order-of-magnitude analysis, the boundary-layer equations are developed. To provide an interface between the analytical partial differentiation notation and the compact tensor notation, a brief review of the most essential theorems of the tensor analysis related to the equations of the fluid dynamics is given. Many useful quantities, such as the contravariant and the covariant metrics and the physical velocity components, are written in both notations

Variational Particle Schemes for the Porous Medium Equation and for the System of Isentropic Euler Equations

Both the porous medium equation and the system of isentropic Euler equations can be considered as steepest descents on suitable manifolds of probability measures in the framework of optimal transport theory. By discretizing these variational characterizations instead of the partial differential equations themselves, we obtain new schemes with remarkable stability properties. We show that they capture successfully the nonlinear features of the flows, such as shocks and rarefaction waves for the isentropic Euler equations. We also show how to design higher order methods for these problems in the optimal transport setting using backward differentiation formula (BDF) multi-step methods or diagonally implicit Runge-Kutta methods.Comment: 36 pages, 9 figures; re-wrote introduction, added 6 references, added discussion of diagonally implicit Runge-Kutta schemes, moved some material to appendice

First derivative-supervised pattern recognition for the flow-injection spectrophotometric discrimination of s-triazines in water

A new spectrophotometric method is presented for the automated differentiation of chloro- (0.5 to 5.0 mg/mL), thio- (0.5 to 5.0 mg/mL) and methoxy-triazines (1 to 10 mg/mL) in water samples. Classification models obtained by K-nearest neighbours, Soft Independent Modeling of Class Analogy, and Partial Least Squares-Discriminatory Analysis were constructed from zero order and first derivative absorption spectra as independent variables, in the spectral range from 210 to 270 nm. Binary responses were used as classifying variables (with/without certain group of triazines). With this dichotomous structure, parameters related to 2x2 contingency tables were used to evaluate the performance of the models. For tap and well water samples, sensitivity and selectivity values equal or higher than 50 % were obtained from autoscaled first derivative spectra, discriminated by Partial Least Squares-Discriminatory Analysis

Computation of bifurcations : Automatic provisioning of variational equations

In the conventional implementations for solving bifurcation problems, Jacobian matrix and its partial derivatives regarding the given problem should be provided manually. This process is not so easy, thus it often induces human errors like computation failures, typing error, especially if the system is higher order. In this paper, we develop a preprocessor that gives Jacobian matrix and partial derivatives symbolically by using SymPy packages on the Python platform. Possibilities about the inclusion of errors are minimized by symbolic derivations and reducing loop structures. It imposes a user only on putting an expression of the equation into a JSON format file. We demonstrate bifurcation calculations for discrete neuron dynamical systems. The system includes an exponential function, which makes the calculation of derivatives complicated, but we show that it can be implemented simply by using symbolic differentiation

Hutchinson Trace Estimation for High-Dimensional and High-Order Physics-Informed Neural Networks

Physics-Informed Neural Networks (PINNs) have proven effective in solving partial differential equations (PDEs), especially when some data are available by seamlessly blending data and physics. However, extending PINNs to high-dimensional and even high-order PDEs encounters significant challenges due to the computational cost associated with automatic differentiation in the residual loss. Herein, we address the limitations of PINNs in handling high-dimensional and high-order PDEs by introducing Hutchinson Trace Estimation (HTE). Starting with the second-order high-dimensional PDEs ubiquitous in scientific computing, HTE transforms the calculation of the entire Hessian matrix into a Hessian vector product (HVP). This approach alleviates the computational bottleneck via Taylor-mode automatic differentiation and significantly reduces memory consumption from the Hessian matrix to HVP. We further showcase HTE's convergence to the original PINN loss and its unbiased behavior under specific conditions. Comparisons with Stochastic Dimension Gradient Descent (SDGD) highlight the distinct advantages of HTE, particularly in scenarios with significant variance among dimensions. We further extend HTE to higher-order and higher-dimensional PDEs, specifically addressing the biharmonic equation. By employing tensor-vector products (TVP), HTE efficiently computes the colossal tensor associated with the fourth-order high-dimensional biharmonic equation, saving memory and enabling rapid computation. The effectiveness of HTE is illustrated through experimental setups, demonstrating comparable convergence rates with SDGD under memory and speed constraints. Additionally, HTE proves valuable in accelerating the Gradient-Enhanced PINN (gPINN) version as well as the Biharmonic equation. Overall, HTE opens up a new capability in scientific machine learning for tackling high-order and high-dimensional PDEs.Comment: Published in Computer Methods in Applied Mechanics and Engineerin