An Energy Approach to the Solution of Partial Differential Equations in Computational Mechanics via Machine Learning: Concepts, Implementation and Applications (1908.10407v2)

Abstract: Partial Differential Equations (PDE) are fundamental to model different phenomena in science and engineering mathematically. Solving them is a crucial step towards a precise knowledge of the behaviour of natural and engineered systems. In general, in order to solve PDEs that represent real systems to an acceptable degree, analytical methods are usually not enough. One has to resort to discretization methods. For engineering problems, probably the best known option is the finite element method (FEM). However, powerful alternatives such as mesh-free methods and Isogeometric Analysis (IGA) are also available. The fundamental idea is to approximate the solution of the PDE by means of functions specifically built to have some desirable properties. In this contribution, we explore Deep Neural Networks (DNNs) as an option for approximation. They have shown impressive results in areas such as visual recognition. DNNs are regarded here as function approximation machines. There is great flexibility to define their structure and important advances in the architecture and the efficiency of the algorithms to implement them make DNNs a very interesting alternative to approximate the solution of a PDE. We concentrate in applications that have an interest for Computational Mechanics. Most contributions that have decided to explore this possibility have adopted a collocation strategy. In this contribution, we concentrate in mechanical problems and analyze the energetic format of the PDE. The energy of a mechanical system seems to be the natural loss function for a machine learning method to approach a mechanical problem. As proofs of concept, we deal with several problems and explore the capabilities of the method for applications in engineering.

• The paper introduces an energy-based loss function where DNNs approximate solutions for PDEs in computational mechanics.
• It validates this approach against benchmark tests in linear elasticity and elastodynamics, with displacement errors as low as 0.5%.
• The paper demonstrates versatility by effectively modeling complex scenarios including fracture, piezoelectricity, and plate bending.

An Energy Approach to the Solution of Partial Differential Equations in Computational Mechanics via Machine Learning: Concepts, Implementation and Applications

Overview

This paper presents a novel approach in computational mechanics that leverages Deep Neural Networks (DNNs) for solving Partial Differential Equations (PDEs) by employing an energy-based methodology. The primary focus is on the application within computational mechanics, particularly for problems that are traditionally addressed using numerical methods like the Finite Element Method (FEM). The authors propose DNNs as a flexible and powerful alternative to conventional discretization techniques.

Methodology

The central concept of this approach is to represent mechanical systems through their energy functions, utilizing DNNs to approximate the solutions to PDEs. The proposed method is built on two core ideas:

1. Energy-Based Loss Function: The energy of the system is used as the loss function in the machine learning framework. This is particularly suited for mechanical problems where the system's energy serves as a natural candidate for the loss function due to its optimization properties.
2. Function Approximation via DNNs: The unknown functions in the PDEs are approximated by the neural network, with its structure defined by various layers and neurons.

The optimization task becomes minimizing the energy-based loss function by adjusting the network parameters (weights and biases). This process is implemented using libraries such as TensorFlow and PyTorch, which provide the necessary computational tools and optimization algorithms.

Implementation

The paper details the implementation strategy of the proposed method:

• Initialization: DNNs are initialized with randomly assigned weights and structured to satisfy boundary conditions.
• Loss Function: The energy of the system, as a function of the neural network outputs, is computed and minimized using stochastic gradient descent and other optimization techniques.
• Training: The neural network is trained on both interior points within the domain and boundary points, where different components of the loss function (e.g., elastic energy, traction forces) are evaluated.
• Validation: The method is validated against several benchmark problems common in computational mechanics, demonstrating its applicability and accuracy.

Applications

Linear Elasticity

For linear elastic problems, the method has been tested on classical problems such as:

• Pressurized Thick Cylinder: Achieving a prediction error of 0.5% in displacement and 3.7% in strain energy.
• Plate with a Circular Hole: Demonstrating a prediction error of 1.8% in displacement and 3.19% in strain energy.
• Hollow Sphere under Internal Pressure: Resulting in a 1.05% error for displacement and 4.44% for strain energy.
• Cube with a Spherical Hole: Completing the test with a 5.3% error in strain energy.

Elastodynamics

The paper extends the approach to time-dependent problems, exemplified by the one-dimensional wave propagation problem. Using space-time discretization, the method displayed relative errors of $1.42 \times 10^{-3}$ in displacement and $5.95 \times 10^{-3}$ in velocity.

Hyperelasticity

A demonstration using a 3D cuboid with Neo-Hookean material properties under torsion applies the method effectively, with detailed results aligning well with traditional FEM simulations.

Fracture Modeling

The phase-field model for fracture is addressed, approximating both the displacement and scalar damage fields. The method successfully simulates crack initialization and propagation in a notched plate under tensile loading.

Piezoelectricity

The method captures the electro-mechanical coupling in piezoelectric materials. A cantilever beam subjected to mechanical and electrical loads validates the method's robustness in handling multi-physical coupling.

Kirchhoff Plate Bending

The method's flexibility is illustrated in solving fourth-order PDEs governing thin plate bending. The deep energy approach, employing autoencoders and tailored activation functions, computes deflections accurately for plates with complex boundary conditions.

Conclusions and Future Work

The paper successfully shows the feasibility of using DNNs as an approximation strategy for solving PDEs in computational mechanics through an energy-based approach. This method offers several advantages over traditional numerical methods, such as direct handling of energy minimization and flexibility in defining complex function spaces.

However, the approach faces challenges due to the non-convex nature of the resulting optimization problems and the inherent complexity of the DNN approximation spaces. Future research should focus on addressing these optimization difficulties and further validating the method's applicability across a wider range of problems and scales.

The integration of advanced machine learning techniques with established principles of computational mechanics opens up new pathways for more efficient and accurate problem-solving in engineering and scientific applications. The paper sets a precedent for further exploration of DNN-based methods in other complex domains, potentially leading to novel computational techniques and enhanced performance in practical applications.