Multilayer neural networks for solving a class of partial differential equations

Solving the initial value problem of ordinary differential equations by Lie group based neural network method

To combine a feedforward neural network (FNN) and Lie group (symmetry) theory of differential equations (DEs), an alternative artificial NN approach is proposed to solve the initial value problems (IVPs) of ordinary DEs (ODEs). Introducing the Lie group expressions of the solution, the trial solution of ODEs is split into two parts. The first part is a solution of other ODEs with initial values of original IVP. This is easily solved using the Lie group and known symbolic or numerical methods without any network parameters (weights and biases). The second part consists of an FNN with adjustable parameters. This is trained using the error back propagation method by minimizing an error (loss) function and updating the parameters. The method significantly reduces the number of the trainable parameters and can more quickly and accurately learn the real solution, compared to the existing similar methods. The numerical method is applied to several cases, including physical oscillation problems. The results have been graphically represented, and some conclusions have been made.

A Dual-Dimer method for training physics-constrained neural networks with minimax architecture

Data sparsity is a common issue to train machine learning tools such as neural networks for engineering and scientific applications, where experiments and simulations are expensive. Recently physics-constrained neural networks (PCNNs) were developed to reduce the required amount of training data. However, the weights of different losses from data and physical constraints are adjusted empirically in PCNNs. In this paper, a new physics-constrained neural network with the minimax architecture (PCNN-MM) is proposed so that the weights of different losses can be adjusted systematically. The training of the PCNN-MM is searching the high-order saddle points of the objective function. A novel saddle point search algorithm called Dual-Dimer method is developed. It is demonstrated that the Dual-Dimer method is computationally more efficient than the gradient descent ascent method for nonconvex-nonconcave functions and provides additional eigenvalue information to verify search results. A heat transfer example also shows that the convergence of PCNN-MMs is faster than that of traditional PCNNs.

Enhancing Function Approximation Abilities of Neural Networks by Training Derivatives

A method to increase the precision of feedforward networks is proposed. It requires prior knowledge of a target's function derivatives of several orders and uses this information in gradient-based training. Forward pass calculates not only the values of the output layer of a network but also their derivatives. The deviations of those derivatives from the target ones are used in an extended cost function, and then, the backward pass calculates the gradient of the extended cost with respect to weights, which is then used by a weights update algorithm. The most accurate approximation is obtained when the training starts with all available derivatives that are then step by step excluded from the extended cost function, starting with the highest orders up until only values are trained. Despite a substantial increase in arithmetic operations per pattern (compared with the conventional training), the method allows to obtain 140-1000 times more accurate approximation for simple cases if the total number of operations is equal. This precision also happens to be out of reach for the regular cost function. The method works well for solving differential equations with neural networks. The cost function is the deviation of the equation's residual from zero, and it can be extended by differentiating the equation itself so no prior information is required. This extension allows to solve 2-D nonlinear partial differential equation 13 times more accurately using seven times fewer grid points. The GPU-efficient algorithm for calculating the gradient of the extended cost function is proposed.

Hermite Functional Link Neural Network for Solving the Van der Pol-Duffing Oscillator Equation

Hermite polynomial-based functional link artificial neural network (FLANN) is proposed here to solve the Van der Pol-Duffing oscillator equation. A single-layer hermite neural network (HeNN) model is used, where a hidden layer is replaced by expansion block of input pattern using Hermite orthogonal polynomials. A feedforward neural network model with the unsupervised error backpropagation principle is used for modifying the network parameters and minimizing the computed error function. The Van der Pol-Duffing and Duffing oscillator equations may not be solved exactly. Here, approximate solutions of these types of equations have been obtained by applying the HeNN model for the first time. Three mathematical example problems and two real-life application problems of Van der Pol-Duffing oscillator equation, extracting the features of early mechanical failure signal and weak signal detection problems, are solved using the proposed HeNN method. HeNN approximate solutions have been compared with results obtained by the well known Runge-Kutta method. Computed results are depicted in term of graphs. After training the HeNN model, we may use it as a black box to get numerical results at any arbitrary point in the domain. Thus, the proposed HeNN method is efficient. The results reveal that this method is reliable and can be applied to other nonlinear problems too.

Transformations of sigma-pi nets: obtaining reflected functions by reflecting weight matrices

This paper presents a methodology that reflected functions by reflecting the weight matrices of an artificial neural network. One of the major problems with the connectionist approach is that trained neural networks can only associate fixed sets of input-output mappings. We provide a methodology which allows the post-trained net to associate different input-output mappings. The different mappings are reflected in a horizontal axis, reflected in a vertical axis and scaling of the initial mapping. The methodology does not train the net on the different mappings but it transforms the weight matrix of the neural network. This paper describes a novel way of utilising sigma-pi neural networks. Our new methodology manipulates sigma-pi unit's weight matrices which transform the unit's output. The weights are cast in a matrix formulation, and then transformations can be performed on the weight matrix of the sigma-pi net. To test the new methodology, the following three steps were carried out on a neural network: (1) the network was trained to perform a mapping function, f; (2) the weights of the network were transformed; and (3) the network was tested to evaluate whether it performs the reflection in the vertical axis,f(ref-vert)(x) = a - f(x). This reflects the function in one dimension. A reflection transformation was used to manipulate the network's weight matrices to obtain a reflection in the vertical axis. Note that the network was not trained to perform the reflection in the vertical axis. The transformation of the weight matrix transformed the function the output performs. This article explains the theory which enables us to perform transformations of sigma-pi networks and obtain reflections of the output by reflecting the weight matrices. These transforms empower the network to perform related mapping tasks once one mapping task has been learnt. This article explains how each transformation is performed and it considers whether a set of 'standard' transformations can indeed be derived.