Research On Solving Partial Differential Equations Based On Physical Information Embedding Neural Network

Partial differential equations(PDEs)are the most basic tools for describing the physical laws of the objective world and play an important role in science and engineering fields.Since most PDEs are difficult to find analytical solutions,numerical methods have been used to obtain approximate solutions of PDEs in the past few decades.However,numerical methods face the curse of dimensionality problem when dealing with high-dimensional PDEs,and it is difficult to efficiently solve inverse problems or parametric PDEs.With the outstanding performance of deep learning in high-dimensional problems such as computer vision,natural language processing,decision control,etc.,deep learning methods have been attempted to solve traditional scientific computing problems,such as molecular simulation,protein structure prediction,weather forecasting,etc。,forming a new type of scientific computing paradigm:artificial intelligence for science(AI4SCI).Using deep learning methods to solve partial differential equations is a very important research direction in this field.Deep learning methods have achieved many successful applications in solving high-dimensional PDEs,inverse problems and parametric PDEs,and significantly improved the efficiency of equation solving.Currently,deep learning methods for PDE can be divided into two categories:data-driven methods and physical information embedding methods.Data-driven methods rely on a large amount of labeled data for PDE solving.When the labeled data is scarce or contains noise,the models trained by these methods have low accuracy,poor generalization ability,and lack interpretability.Physical information embedding neural network methods can reduce the dependence on labeled data,and can be further divided into two categories according to the embedding method.The first category is physics-driven method,which uses implicit embedding method to constrain neural network by using PDE’s governing equation,initial conditions and boundary conditions as loss function terms.This method can not rely on predefined grids and does not need labeled data at all,but it is currently difficult to deal with PDEs with a point source and has low efficiency in solving parametric PDEs.The second category is data-mechanism fusion method,which uses explicit embedding method to encode physics information as part of network architecture in some way.This method is superior to data-driven methods in prediction accuracy and extrapolation ability,but there are problems such as poor generality and weak ability to deal with complex nonlinear terms and unknown terms.This thesis explores methods for solving PDEs using neural networks with embedded physical information,addressing the challenge of limited availability of labeled data in practical applications.The approach enables efficient and accurate PDE solving,even with little or no labeled data.This thesis focuses on two types of problems:solving individual equation instances and solving parametric PDEs.Effective methods for embedding physical information.into neural networks and deep learning optimization techniques are investigated,addressing three key issues in the field.The main contributions and innovations of this thesis are as follows:1.This thesis proposes a general Physics-Informed Neural Networks(PINNs)method for solving PDEs with a point source.By approximating the Dirac δ function with a symmetric unimodal probability density function,the singularity problem associated with the point source is eliminated.A lower-bound constrained uncertainty weighting method is introduced to balance multiple loss function terms with large numerical differences.Additionally,a multi-scale deep neural network architecture that incorporates Sine activation functions is proposed to enhance its expressive power.Experiments on multiple PDEs with point sources demonstrate that this method significantly outperforms existing deep learning methods in terms of model prediction accuracy and training convergence speed.2.This thesis proposes Meta-Auto-Decoder(MAD)to address the issue of low efficiency when solving parametric PDEs using physics-driven methods.From a meta-learning perspective,the solution of equation instances corresponding to different PDE parameters is viewed as different tasks.By combining pre-training and fine-tuning,the cost of retraining the model is reduced,enabling efficient solutions of parametric PDEs in the absence of labeled data.The principle of MAD is explained from the perspective of manifold learning.Experiments on multiple parametric PDEs show that MAD can significantly improve fine-tuning speed while maintaining prediction accuracy.3.This thesis proposes PDE-Net++,a data-mechanism fusion method that addresses the issues of poor universality and weak ability to handle complex nonlinear and unknown terms.By combining finite difference methods and black box models,physical information is explicitly embedded in the neural network architecture.Two new types of trainable differential operators are proposed to improve the numerical stability of long time series prediction,solving the problem of poor accuracy of black box models in predicting the evolution of complex spatiotemporal dynamic systems.Numerous numerical experiments have shown that PDENet++has higher prediction accuracy compared to black box models.By implicitly or explicitly embedding physical information into neural networks,this thesis improves the existing deep learning methods and conducts a large number of numerical experiments,demonstrating the feasibility of efficiently solving PDEs based on deep learning.