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At least 163 records · Page 9
All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

HomPINNs: Homotopy physics-informed neural networks for learning multiple solutions of nonlinear elliptic differential equations

Physics-informed neural networks (PINNs) based machine learning is an emerging framework for solving nonlinear differential equations. However, due to the implicit regularity of neural network structure, PINNs can only find the flattest solution in most cases by minimizing the loss functions. In this paper, we combine PINNs with the homotopy continuation method, a classical numerical method to compute isolated roots of polynomial systems, and propose a new deep learning framework, named homotopy physics-informed neural networks (HomPINNs), for solving multiple solutions of nonlinear elliptic differential equations. The implementation of an HomPINN is a homotopy process that is composed of the training of a fully connected neural network, named the starting neural network, and training processes of several PINNs with different tracking parameters. The starting neural network is to approximate a starting function constructed by the trivial solutions, while other PINNs are to minimize the loss functions defined by boundary condition and homotopy functions, varying with different tracking parameters. These training processes are regraded as different steps of a homotopy process, and a PINN is initialized by the well-trained neural network of the previous step, while the first starting neural network is initialized using the default initialization method. Finally, several numerical examples are presented to show the efficiency of our proposed HomPINNs, including reaction-diffusion equations with a heart-shaped domain.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1982

Propagation of local errors in the solutions of the differential equations for orbital elements

The propagation of errors in the solutions of the differential equations for the orbital elements of perturbed two-body motion is investigated. It is shown that the error in the time-element grows linearly for differential equations for orbital elements when only perturbations are present on the right-hand side, cubically for formulations which have a two-body term on the right-hand side, and linearly for formulations based upon extended phase space Hamiltonians.

Bond, V. R.↗
DOE OSTI2025

Reduced basis approximations of parameterized dynamical partial differential equations via neural networks

Projection-based reduced order models are effective at approximating parameter-dependent differential equations that are parametrically separable. When parametric separability is not satisfied, which occurs in both linear and nonlinear problems, projection-based methods fail to adequately reduce the computational complexity. Devising alternative reduced order models is crucial for obtaining efficient and accurate approximations to expensive high-fidelity models. In this work, we develop a timestepping procedure for dynamical parameter-dependent problems, in which a neural-network is trained to propagate the coefficients of a reduced basis expansion. This results in an online stage with a computational cost independent of the size of the underlying problem. Here, we demonstrate our method on several parabolic partial differential equations, including a problem that is not parametrically separable.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1971

Similarity and generalized finite-difference solutions of parabolic partial differential equations.

Techniques are presented for obtaining generalized finite-difference solutions to partial differential equations of the parabolic type. It is shown that the advantages of similarity in the solution of similar problems are generally not lost if the solution to the original partial differential equations is effected in the physical plane by finite-difference methods. The analysis results in a considerable saving in computational effort in the solution of both similar and nonsimilar problems. Several examples, including both the heat-conduction equation and the boundary-layer equations, are given. The analysis also provides a practical means of estimating the accuracy of finite-difference solutions to parabolic equations.

Clausing, A. M.↗
NASA NTRS1973

Program for solution of ordinary differential equations

A program for the solution of linear and nonlinear first order ordinary differential equations is described and user instructions are included. The program contains a new integration algorithm for the solution of initial value problems which is particularly efficient for the solution of differential equations with a wide range of eigenvalues. The program in its present form handles up to ten state variables, but expansion to handle up to fifty state variables is being investigated.

Sloate, H.↗
NASA NTRS1988

Minimal parameter solution of the orthogonal matrix differential equation

As demonstrated in this work, all orthogonal matrices solve a first order differential equation. The straightforward solution of this equation requires n sup 2 integrations to obtain the element of the nth order matrix. There are, however, only n(n-1)/2 independent parameters which determine an orthogonal matrix. The questions of choosing them, finding their differential equation and expressing the orthogonal matrix in terms of these parameters are considered. Several possibilities which are based on attitude determination in three dimensions are examined. It is shown that not all 3-D methods have useful extensions to higher dimensions. It is also shown why the rate of change of the matrix elements, which are the elements of the angular rate vector in 3-D, are the elements of a tensor of the second rank (dyadic) in spaces other than three dimensional. It is proven that the 3-D Gibbs vector (or Cayley Parameters) are extendable to other dimensions. An algorithm is developed employing the resulting parameters, which are termed Extended Rodrigues Parameters, and numerical results are presented of the application of the algorithm to a fourth order matrix.

Bar-Itzhack, Itzhack Y.↗
NASA NTRS1988

Minimal parameter solution of the orthogonal matrix differential equation

As demonstrated in this work, all orthogonal matrices solve a first order differential equation. The straightforward solution of this equation requires n sup 2 integrations to obtain the element of the nth order matrix. There are, however, only n(n-1)/2 independent parameters which determine an orthogonal matrix. The questions of choosing them, finding their differential equation and expressing the orthogonal matrix in terms of these parameters are considered. Several possibilities which are based on attitude determination in three dimensions are examined. It is shown that not all 3-D methods have useful extensions to higher dimensions. It is also shown why the rate of change of the matrix elements, which are the elements of the angular rate vector in 3-D, are the elements of a tensor of the second rank (dyadic) in spaces other than three dimensional. It is proven that the 3-D Gibbs vector (or Cayley Parameters) are extendable to other dimensions. An algorithm is developed employing the resulting parameters, which are termed Extended Rodrigues Parameters, and numerical results are presented of the application of the algorithm to a fourth order matrix.

Baritzhack, Itzhack Y.↗
NASA NTRS1990

Minimal parameter solution of the orthogonal matrix differential equation

As demonstrated in this work, all orthogonal matrices solve a first order differential equation. The straightforward solution of this equation requires n sup 2 integrations to obtain the element of the nth order matrix. There are, however, only n(n-1)/2 independent parameters which determine an orthogonal matrix. The questions of choosing them, finding their differential equation and expressing the orthogonal matrix in terms of these parameters are considered. Several possibilities which are based on attitude determination in three dimensions are examined. It is shown that not all 3-D methods have useful extensions to higher dimensions. It is also shown why the rate of change of the matrix elements, which are the elements of the angular rate vector in 3-D, are the elements of a tensor of the second rank (dyadic) in spaces other than three dimensional. It is proven that the 3-D Gibbs vector (or Cayley Parameters) are extendable to other dimensions. An algorithm is developed emplying the resulting parameters, which are termed Extended Rodrigues Parameters, and numerical results are presented of the application of the algorithm to a fourth order matrix.

Bar-Itzhack, Itzhack Y.↗
NASA NTRS1988

Real-time optical laboratory solution of parabolic differential equations

An optical laboratory matrix-vector processor is used to solve parabolic differential equations (the transient diffusion equation with two space variables and time) by an explicit algorithm. This includes optical matrix-vector nonbase-2 encoded laboratory data, the combination of nonbase-2 and frequency-multiplexed data on such processors, a high-accuracy optical laboratory solution of a partial differential equation, new data partitioning techniques, and a discussion of a multiprocessor optical matrix-vector architecture.

Casasent, David↗
DOE OSTI2023

Group Projected subspace pursuit for IDENTification of variable coefficient differential equations (GP-IDENT)

We propose an effective and robust algorithm for identifying partial differential equations (PDEs) with space-time varying coefficients from the noisy observation of a single solution trajectory. Identifying unknown differential equations from noisy data is a difficult task, and it is even more challenging with space and time varying coefficients in the PDE. The proposed algorithm, GP-IDENT, has three ingredients: (i) we use B-spline bases to express the unknown space and time varying coefficients, (ii) we propose Group Projected Subspace Pursuit (GPSP) to find a sequence of candidate PDEs with various levels of complexity, and (iii) we propose a new criterion for model selection using the Reduction in Residual (RR) to choose an optimal one among a pool of candidates. The new GPSP considers group projected subspaces which is more robust than existing methods in distinguishing correlated group features. We test GP-IDENT on a variety of PDEs and PDE systems, and compare it with the state-of-the-art parametric PDE identification algorithms under different settings to illustrate its outstanding performance. Furthermore, our experiments show that GP-IDENT is effective in identifying the correct terms from a large dictionary, and our model selection scheme is robust to noise.

Data-driven method↗
NASA NTRS1974

Multi-off-grid methods in multi-step integration of ordinary differential equations

Description of methods of solving first- and second-order systems of differential equations in which all derivatives are evaluated at off-grid locations in order to circumvent the Dahlquist stability limitation on the order of on-grid methods. The proposed multi-off-grid methods require off-grid state predictors for the evaluation of the n derivatives at each step. Progressing forward in time, the off-grid states are predicted using a linear combination of back on-grid state values and off-grid derivative evaluations. A comparison is made between the proposed multi-off-grid methods and the corresponding Adams and Cowell on-grid integration techniques in integrating systems of ordinary differential equations, showing a significant reduction in the error at larger step sizes in the case of the multi-off-grid integrator.

Beaudet, P. R.↗