Jiayi Li, Guancheng Zhou, Yisi Luo, Ruihua Chen, Penghao Jiang, Deyu Meng, Flora D. Salim, Piotr Koniusz
Large-scale PDE pre-training trains a single model on the data from multiple partial differential equations (PDEs). However, adding heterogeneous PDEs with different governing equations, physical variables, and variable interactions does not always improve predictive performance across tasks. Existing PDE mixture-of-experts models mitigate cross-task interference through learned routing, but typically use homogeneous expert architectures, making it difficult to directly reflect differences in state evolution across equations. We introduce HotPOT, an operator transformer with heterogeneous physics experts for large-scale PDE pre-training. HotPOT uses a shared encoder to capture information common across PDEs, while each PDE family uses a neural-numerical expert that pairs a learnable predictor with a fixed numerical update containing no trainable parameters. Each predictor outputs the quantities required by its paired numerical update, which then computes the next state. This design allows different PDEs to share a common encoder while retaining their own learnable predictors and numerical updates. Experiments across five PDE families and six settings show that HotPOT outperforms MoE-POT in all 24 matched-scale comparisons before and after fine-tuning, reducing rollout error by 50.7% on average. Notably, the 17M model also surpasses substantially larger MoE-POT models, and the performance gains extend to zero-shot prediction under unseen physical conditions. Code is available at https://github.com/cruiseresearchgroup/HotPOT-code.
Jiayi Li, Penghao Jiang, Baiyu Chen, Yifan Duan, Hira Saleem, Zhaonan Wang, Piotr Koniusz, Flora D. Salim
Neural operators efficiently simulate dynamical systems by repeatedly predicting the next state. However, small errors at each step can accumulate, causing long-term predictions to drift from the true trajectory. Long-term error reflects both the propagation of earlier prediction errors and new errors introduced at each step. How to reduce these errors is therefore a critical question. We propose the Spectral Generator Neural Operator (SGNO), which combines a linear estimate of the next state with a learned correction. The linear component uses consecutive states in the training data to fit a map from the current state to the next. A neural network corrects this prediction by learning changes that the linear model does not capture, including nonlinear interactions. Both components are trained jointly toward the same next-state target. Experiments across 13 periodic PDE configurations in one to three dimensions show that SGNO accurately learns state evolution on linear tasks and reduces 100-step rollout error by 13-74% over the strongest baseline under the main comparison protocol on each nonlinear task. Longer-horizon extrapolation tests show lower late-window prediction errors on several nonlinear tasks. These results show that combining fitted propagation with learned state-dependent correction can improve long-horizon autoregressive prediction on the evaluated PDE tasks. Code is available at https://github.com/cruiseresearchgroup/SGNO.