Interpolating Neural Operator (INO): A Data-Free and Efficient Approach for Learning PDE Solution Operators

2026-09-29

AI Summary

Existing neural operators typically require massive datasets and expensive GPU training, while offering limited means to assess single-prediction accuracy. This work proposes the Interpolative Network Operator (INO), a data-free framework grounded in the weak form of partial differential equations. By integrating Karhunen–Loève coordinates, C-HiDeNN subnetworks, and greedy alternating least squares, INO achieves millisecond-level inference on a single CPU core. Furthermore, it establishes an error-bound estimation mechanism that operates without reference solutions, enabling input-range validation and extrapolation handling. Experimental results demonstrate that INO outperforms PINN and DeepONet on most benchmarks by up to 53× in accuracy, while achieving an 80× speedup in training compared to GPU-based baselines.

Abstract

Neural operators have become a popular approach to approximate the solution operators of parametric partial differential equations (PDEs). However, existing neural operators either require a large amount of simulation data or a long physics-informed training on GPUs, and they cannot tell how accurate an individual prediction is. In this paper, we propose the Interpolating Neural Operator (INO), a data-free interpolating neural network that is trained directly on the weak form of the PDE. In INO, the Karhunen-Loève coordinates of the input field are treated as additional inputs together with the spatial coordinates, and each input is approximated by a C-HiDeNN sub-network whose trainable parameters are nodal values. Since the network is multilinear in its parameters, training reduces to a sequence of one-dimensional linear solves by greedy alternating least squares. As a result, INO trains on one CPU core and predicts a new solution in microseconds. For coercive problems, the total error of every prediction is bounded by a computable residual bound that requires no reference solution, and the same bound applies to the predictions of other methods that satisfy the boundary conditions exactly. Before each prediction, INO checks whether the leading coordinates of the input lie within the range on which it is trained, and inputs outside this range can be passed to a conventional solver or to an INO trained on a wider range. INO is compared with physics-informed FNO and DeepONet on different benchmarks. INO is the most accurate model on most of these problems, by 15$\times$ on two-dimensional Helmholtz at $65^2$ and 53$\times$ on the diffusion-reaction benchmark, and on the one- and two-dimensional problems its training on one CPU core takes 3-80$\times$ less time than the physics-informed baselines on one GPU.

Problem

• Neural Operators
• Parametric PDEs
• Data-Free Learning
• Error Estimation
• Solution Operators

Innovation

• Interpolating Neural Operator
• data-free learning
• Karhunen-Loève expansion
• greedy alternating least squares
• computable residual bound