Latent Neural Operator for Solving Forward and Inverse PDE Problems

TL;DR

The paper introduces the Latent Neural Operator (LNO), a novel approach that performs PDE solving in the latent space to improve computational efficiency and accuracy, enabling effective interpolation and extrapolation for forward and inverse problems.

Contribution

LNO is the first neural operator to operate in the latent space, reducing memory and computation while maintaining high accuracy for PDE tasks.

Findings

LNO reduces GPU memory usage by 50%.

LNO speeds up training by 1.8 times.

LNO achieves state-of-the-art accuracy on multiple benchmarks.

Abstract

Neural operators effectively solve PDE problems from data without knowing the explicit equations, which learn the map from the input sequences of observed samples to the predicted values. Most existing works build the model in the original geometric space, leading to high computational costs when the number of sample points is large. We present the Latent Neural Operator (LNO) solving PDEs in the latent space. In particular, we first propose Physics-Cross-Attention (PhCA) transforming representation from the geometric space to the latent space, then learn the operator in the latent space, and finally recover the real-world geometric space via the inverse PhCA map. Our model retains flexibility that can decode values in any position not limited to locations defined in the training set, and therefore can naturally perform interpolation and extrapolation tasks particularly useful for inverse problems. Moreover, the proposed LNO improves both prediction accuracy and computational efficiency. Experiments show that LNO reduces the GPU memory by 50%, speeds up training 1.8 times, and reaches state-of-the-art accuracy on four out of six benchmarks for forward problems and a benchmark for inverse problem. Code is available at https://github.com/L-I-M-I-T/LatentNeuralOperator.