Invariant deep neural networks under the finite group for solving partial differential equations

Utilizing physics-informed neural networks (PINN) to solve partial differential equations (PDEs) becomes a hot issue and also shows its great powers, but still suffers from the dilemmas of limited predicted accuracy in the sampling domain and poor prediction ability beyond the sampling domain which are usually mitigated by adding the physical properties of PDEs into the loss function or by employing smart techniques to change the form of loss function for special PDEs. In this paper, we design a symmetry-enhanced deep neural network (sDNN) which makes the architecture of neural networks invariant under the finite group through expanding the dimensions of weight matrixes and bias vectors in each hidden layers by the order of finite group if the group has matrix representations, otherwise extending the set of input data and the hidden layers except for the first hidden layer by the order of finite group. However, the total number of training parameters is only about one over the order of finite group of the original PINN size due to the symmetric architecture of sDNN. Furthermore, we give special forms of weight matrixes and bias vectors of sDNN, and rigorously prove that the architecture itself is invariant under the finite group and the sDNN has the universal approximation ability to learn the function keeping the finite group. Numerical results show that the sDNN has strong predicted abilities in and beyond the sampling domain and performs far better than the vanilla PINN with fewer training points and simpler architecture.

Symmetry group based domain decomposition to enhance physics-informed neural networks for solving partial differential equations

Domain decomposition provides an effective way to tackle the dilemma of physics-informed neural networks (PINN) which struggle to accurately and efficiently solve partial differential equations (PDEs) in the whole domain, but the lack of efficient tools for dealing with the interfaces between two adjacent sub-domains heavily hinders the training effects, even leads to the discontinuity of the learned solutions. In this paper, we propose a symmetry group based domain decomposition strategy to enhance the PINN for solving the forward and inverse problems of the PDEs possessing a Lie symmetry group. Specifically, for the forward problem, we first deploy the symmetry group to generate the dividing-lines having known solution information which can be adjusted flexibly and are used to divide the whole training domain into a finite number of non-overlapping sub-domains, then utilize the PINN and the symmetry-enhanced PINN methods to learn the solutions in each sub-domain and finally stitch them to the overall solution of PDEs. For the inverse problem, we first utilize the symmetry group acting on the data of the initial and boundary conditions to generate labeled data in the interior domain of PDEs and then find the undetermined parameters as well as the solution by only training the neural networks in a sub-domain. Consequently, the proposed method can predict high-accuracy solutions of PDEs which are failed by the vanilla PINN in the whole domain and the extended physics-informed neural network in the same sub-domains. Numerical results of the Korteweg-de Vries equation with a translation symmetry and the nonlinear viscous fluid equation with a scaling symmetry show that the accuracies of the learned solutions are improved largely.

Enforcing Continuous Physical Symmetries in Deep Learning Network for Solving Partial Differential Equations

As a typical {application} of deep learning, physics-informed neural network (PINN) {has been} successfully used to find numerical solutions of partial differential equations (PDEs), but how to improve the limited accuracy is still a great challenge for PINN. In this work, we introduce a new method, symmetry-enhanced physics informed neural network (SPINN) where the invariant surface conditions induced by the Lie symmetries of PDEs are embedded into the loss function of PINN, for improving the accuracy of PINN. We test the effectiveness of SPINN via two groups of ten independent numerical experiments for the heat equation, Korteweg-de Vries (KdV) equation and potential Burgers {equations} respectively, which shows that SPINN performs better than PINN with fewer training points and simpler architecture of neural network. Furthermore, we discuss the computational overhead of SPINN in terms of the relative computational cost to PINN and show that the training time of SPINN has no obvious increases, even less than PINN for some cases.