A Physics-informed Multi-resolution Neural Operator

TL;DR

This work tackles operator learning for parametric PDEs when input data are unevenly discretized by introducing a physics-informed, data-free framework (PI-RINO). It first embeds irregular input realizations into a fixed latent space via a dictionary of pre-trained INR-based basis functions, yielding embedding coefficients $\boldsymbol{\alpha}(\boldsymbol{u})$, and then uses a simple MLP to map $(\boldsymbol{\alpha}(\boldsymbol{u}), \boldsymbol{y})$ to the solution $s_{\boldsymbol{\theta}}(\boldsymbol{y})$, with the governing PDE enforced through a finite-difference physics loss $\mathcal{L} = \mathcal{L}_{\text{pde}} + \mathcal{L}_{\text{bc}} + \mathcal{L}_{\text{ic}}$ and input reconstruction $\tilde{u}(\boldsymbol{y}) = \sum_l \psi_l(\boldsymbol{y}) \alpha_l$. Compared to Autodiff-based PINNs, the FD-based enforcement yields comparable accuracy while delivering substantial speedups (roughly $2\times$ for 1D and up to $10\times$ for 2D problems) across 1D antiderivative, 2D heat conduction, and Biot consolidation benchmarks. The approach handles multi-resolution, misaligned inputs and sparse data, and shows promise for extension to geometries, multi-fidelity, and multi-modal data.

Abstract

The predictive accuracy of operator learning frameworks depends on the quality and quantity of available training data (input-output function pairs), often requiring substantial amounts of high-fidelity data, which can be challenging to obtain in some real-world engineering applications. These datasets may be unevenly discretized from one realization to another, with the grid resolution varying across samples. In this study, we introduce a physics-informed operator learning approach by extending the Resolution Independent Neural Operator (RINO) framework to a fully data-free setup, addressing both challenges simultaneously. Here, the arbitrarily (but sufficiently finely) discretized input functions are projected onto a latent embedding space (i.e., a vector space of finite dimensions), using pre-trained basis functions. The operator associated with the underlying partial differential equations (PDEs) is then approximated by a simple multi-layer perceptron (MLP), which takes as input a latent code along with spatiotemporal coordinates to produce the solution in the physical space. The PDEs are enforced via a finite difference solver in the physical space. The validation and performance of the proposed method are benchmarked on several numerical examples with multi-resolution data, where input functions are sampled at varying resolutions, including both coarse and fine discretizations.

Figures (14)

• Figure 1: Illustration of a single-resolution dataset that typical neural operators use (left), compared to the multi-resolution dataset used in this study (right). The single-resolution dataset requires the input functions to be sampled at consistent location across all the realizations, while in the multi-resolution dataset does not have that requirement, as the input functions can be arbitrarily sampled across all the realizations.
• Figure 2: Schematic of the architecture of Physics-informed Resolution Independent Neural Operator (PI-RINO) for approximating the operator $\mathcal{G}:u \mapsto s$, following the PDE: $\mathcal{N}(u,s)=0$.
• Figure 3: The physics loss enforced by convoluting a finite difference stencil across the entire grid of structured collocation points in the domain. This figure demonstrates the FD loss term computed for the $(i)$-th sample at the $(p,q)$-th collocation point for the representative PDE: $\Delta^2 s=u$. The final physics loss for the $(i)$-th sample is computed by taking the sum of all the residuals/losses across all the collocation points.
• Figure 4: Antiderivative Example: Convergence profiles (MSE vs. Epochs) for the training and test sets for three different activation functions when the initial conditions are hard/soft constrained.
• Figure 5: Antiderivative Example: of output function prediction errors (in log rel. MSE) for the training and testing datasets after training for (a) Mish, (b) Tanh, and (c) ReLU activation functions when the initial condition was hard constrained during training.

Theorems & Definitions (1)

• Remark 1