A foundational neural operator that continuously learns without forgetting

Machine learning has witnessed substantial growth, leading to the development of advanced artificial intelligence models crafted to address a wide range of real-world challenges spanning various domains, such as computer vision, natural language processing, and scientific computing. Nevertheless, the creation of custom models for each new task remains a resource-intensive undertaking, demanding considerable computational time and memory resources. In this study, we introduce the concept of the Neural Combinatorial Wavelet Neural Operator (NCWNO) as a foundational model for scientific computing. This model is specifically designed to excel in learning from a diverse spectrum of physics and continuously adapt to the solution operators associated with parametric partial differential equations (PDEs). The NCWNO leverages a gated structure that employs local wavelet experts to acquire shared features across multiple physical systems, complemented by a memory-based ensembling approach among these local wavelet experts. This combination enables rapid adaptation to new challenges. The proposed foundational model offers two key advantages: (i) it can simultaneously learn solution operators for multiple parametric PDEs, and (ii) it can swiftly generalize to new parametric PDEs with minimal fine-tuning. The proposed NCWNO is the first foundational operator learning algorithm distinguished by its (i) robustness against catastrophic forgetting, (ii) the maintenance of positive transfer for new parametric PDEs, and (iii) the facilitation of knowledge transfer across dissimilar tasks. Through an extensive set of benchmark examples, we demonstrate that the NCWNO can outperform task-specific baseline operator learning frameworks with minimal hyperparameter tuning at the prediction stage. We also show that with minimal fine-tuning, the NCWNO performs accurate combinatorial learning of new parametric PDEs.