A Robust Foundation Model for Conservation Laws: Injecting Context into Flux Neural Operators via Recurrent Vision Transformers
For researchers in scientific machine learning, this work provides a method to solve conservation laws without knowing the governing equation, though it is an incremental improvement over existing Flux NO.
The paper introduces a hypernetwork architecture that augments Flux Neural Operators with recurrent Vision Transformers to infer and solve conservation laws without explicit PDE coefficients, achieving robust generalization across unseen flux systems while preserving long-time prediction accuracy.
We propose an architecture that augments the Flux Neural Operator (Flux NO), which combines the classical finite volume method (FVM) with neural operators, with ViT-based context injection. Our model is formulated as a hypernetwork: it extracts solution dynamics over a finite temporal window, encodes them with a recurrent Vision Transformer, and generates the parameters of a context-conditioned neural operator. This enables the model to infer and solve conservation laws without explicit access to the governing equation or PDE coefficients. Experimentally, we show that the proposed method preserves the robustness, generalization ability, and long-time prediction advantages of Flux NO over standard neural operators, while delivering reliable numerical solutions across a broad range of conservative systems, including previously unseen fluxes. Our code is available at https://github.com/xx257xx/CONTEXT_FLUX_NO.