fPINNs: Fractional physics-informed neural networks

Analysis

Why This Paper Matters

Fractional partial differential equations (PDEs) are increasingly used to model complex phenomena with memory and non-local interactions, such as anomalous diffusion in porous media, viscoelastic materials, and financial systems. Traditional numerical solvers for fractional PDEs are often computationally expensive and require careful handling of non-local operators. This paper introduces fPINNs, a novel extension of physics-informed neural networks (PINNs) that directly incorporates fractional derivatives into the neural network training process. By doing so, it offers a mesh-free, flexible alternative to classical numerical methods, potentially making fractional PDEs more accessible to practitioners in science and engineering.

The significance of this work lies in its demonstration that neural networks can be trained to satisfy fractional differential operators, which are inherently non-local and challenging to handle. This opens up new possibilities for solving inverse problems and parameter estimation in fractional models, which are often ill-posed and difficult with traditional techniques. Moreover, the fPINN framework is generalizable to different definitions of fractional derivatives, making it a versatile tool for a wide range of applications.

Technical Contributions

• Fractional derivative computation: The paper proposes a method to compute fractional derivatives within the neural network's loss function using numerical discretization, such as the Grünwald-Letnikov or Caputo definitions, enabling gradient-based optimization.
• Loss function design: The loss function includes both the residual of the fractional PDE and boundary/initial conditions, allowing the network to learn solutions that satisfy the physical constraints.
• Mesh-free approach: Unlike traditional solvers, fPINNs do not require a structured grid, making them suitable for complex geometries and high-dimensional problems.
• Flexibility: The framework can be adapted to various fractional orders and derivative definitions, providing a unified approach for different types of fractional PDEs.

Results

The paper demonstrates the effectiveness of fPINNs on several benchmark problems, including fractional diffusion and advection-diffusion equations. The reported results show that fPINNs achieve accuracy comparable to traditional numerical methods, with relative errors in the range of 10^-3 to 10^-4 for the tested cases. The convergence behavior is consistent with increasing network size and training data points. However, the paper does not provide extensive comparisons with state-of-the-art fractional PDE solvers, and the computational cost is not thoroughly analyzed.

Significance

The introduction of fPINNs marks a significant step forward in the application of physics-informed neural networks to a broader class of differential equations. By enabling the solution of fractional PDEs, this work expands the scope of PINNs to areas where non-local and memory effects are crucial. The mesh-free nature of the approach could be particularly beneficial for problems in irregular domains or where data is sparse. Furthermore, the ability to handle fractional derivatives opens up new avenues for solving inverse problems, such as parameter identification in fractional models, which are common in real-world applications. As the field of scientific machine learning grows, fPINNs contribute to the toolkit of researchers and engineers, potentially leading to more efficient and flexible simulations of complex physical systems.