Numerical Solution of Mixed-Dimensional PDEs Using a Neural Preconditioner

TL;DR

The paper addresses the ill-conditioning of mixed-dimensional PDEs by introducing an unsupervised, neural, nonlinear preconditioner that operates in a matrix-free fashion. It learns a map \mathscr{P}(v,\\mu) that approximates the action of the inverse on selected right-hand sides, enabling faster GMRES convergence for a 3D-1D coupled system. Key innovations include using a U-Net-based preconditioner with dual inputs (rhs-like vector and 1D distance descriptor), data augmentation via Krylov and random high-frequency vectors, and an unsupervised risk that does not require solving the full linear systems during training. Numerical experiments demonstrate improved convergence over conventional preconditioners, effective generalization across 1D topology changes and mesh resolutions, and promising scalability under ensemble computations, while highlighting limitations for very large-scale problems and tensor-product domain restrictions.

Abstract

Mixed-dimensional partial differential equations (PDEs) are characterized by coupled operators defined on domains of varying dimensions and pose significant computational challenges due to their inherent ill-conditioning. Moreover, the computational workload increases considerably when attempting to accurately capture the behavior of the system under significant variations or uncertainties in the low-dimensional structures such as fractures, fibers, or vascular networks, due to the inevitable necessity of running multiple simulations. In this work, we present a novel preconditioning strategy that leverages neural networks and unsupervised operator learning to design an efficient preconditioner specifically tailored to a class of 3D-1D mixed-dimensional PDEs. The proposed approach is capable of generalizing to varying shapes of the 1D manifold without retraining, making it robust to changes in the 1D graph topology. Moreover, thanks to convolutional neural networks, the neural preconditioner can adapt over a range of increasing mesh resolutions of the discrete problem, enabling us to train it on low resolution problems and deploy it on higher resolutions. Numerical experiments validate the effectiveness of the preconditioner in accelerating convergence in iterative solvers, demonstrating its appeal and limitations over traditional methods. This study lays the groundwork for applying neural network-based preconditioning techniques to a broader range of coupled multi-physics systems.

Figures (7)

• Figure 1: Solution of a 3D-1D coupled problem for a fixed graph geometry $\Lambda$. Left: plot of 1D solution $u_\Lambda$. Center: slices of 3D solution $u_\Omega$. Right: slices plot of the distance function $d(\Lambda)$ associated to the graph.
• Figure 2: Pictorial summary for the construction of the augmented training set $\mathcal{K}^\mu$. The data augmentation set $\mathcal{D}^{\mu_j}$ contains vectors in the unit sphere with the desired frequency content. Here, $\mathcal{B}/||\mathcal{B}||:=\left\{b/||b||: b \in \mathcal{B}\right\} \subset \mathbb{S}^{N_h-1}$ is the set of normalized right-hand side vectors.
• Figure 3: Schematic representation of the U-Net architecture $\mathcal{U}_3$. Tensor data are represented by blocks, where the number of channels corresponds to the depth of the blocks. The tensor shape, $c \times n_1 \times n_2 \times n_3$, is reported on the top of the blocks. Solid line arrows represent layer action, while dashed arrows represent channel-staking skip connection.
• Figure 4: Examples of the graphs $\Lambda$ considered in the numerical tests, with increasing geometrical complexity.
• Figure 5: Convergence of train and test relative errors of the U-Net $\mathcal{U}_3^\star$ when trained on different datasets, for the physical constants $k_\Omega = 10^{-3}$, $\sigma_\Omega=10^{-3}$.

Theorems & Definitions (8)

• Remark 1
• Remark 2
• Remark 3
• Lemma 1
• proof
• Remark 4
• Lemma A.1
• proof