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.

翻译：神经算子无需显式方程即可从数据中有效求解偏微分方程问题，其通过学习从观测样本的输入序列到预测值的映射关系。现有研究大多在原始几何空间中构建模型，当样本点数量较大时会导致高昂的计算成本。本文提出在潜空间中求解偏微分方程的潜神经算子。具体而言，我们首先提出物理交叉注意力机制，将表征从几何空间转换至潜空间，随后在潜空间中学习算子，最终通过逆物理交叉注意力映射恢复真实几何空间。该模型保留了可解码任意位置数值的灵活性（不限于训练集定义的位置），因此能自然执行插值与外推任务，这对反问题求解尤为有益。此外，所提出的潜神经算子同时提升了预测精度与计算效率。实验表明，潜神经算子可降低50%的GPU内存占用，加速训练1.8倍，并在六个正问题基准中的四个及一个反问题基准上达到最优精度。代码发布于https://github.com/L-I-M-I-T/LatentNeuralOperator。