Conditioning on PDE Parameters to Generalise Deep Learning Emulation of Stochastic and Chaotic Dynamics

The paper clearly specifies and motivates a parametric local‑attention transformer with circular padding and learnable relative positional encodings, and it repeatedly asserts that these choices preserve translation equivariance in periodic domains; however, it offers no formal proofs or explicit theorems beyond empirical demonstrations and architectural descriptions (e.g., local attention with relative PE and circular padding preserves translation equivariance; see methodology and appendix discussions). The candidate solution provides formal statements and sketches for (A) translation equivariance, (B) uniform approximation over parameters via partition‑of‑unity gating, (C) autoregressive stability bounds, and (D) Wasserstein‑based perturbation bounds for a probabilistic emulator. But parts (B)–(D) rely on substantial additional assumptions not present in the paper (e.g., a uniform-in-grid-size universal approximation property for local attention, explicit Lipschitz and forward-invariance conditions, and uniform Wasserstein contractivity of both true and learned kernels). Moreover, the ‘parallel experts’ construction in (B) needs either enough attention heads or an explicit pre‑attention gating to avoid softmax coupling; this key architectural caveat is unstated. Thus, the paper is empirically sound but theoretically incomplete, and the model’s solution is mathematically plausible yet incomplete without these missing hypotheses and architectural clarifications. Key paper passages supporting this reading include their local-attention design with circular padding and relative positional encodings claimed to maintain translation equivariance, and the AdaLN conditioning mechanism for parametric generalization, all presented without rigorous proofs and primarily validated empirically . The long-horizon stability and statistical fidelity are also shown empirically (e.g., PSD and PDF matches for the beta-plane system), not via bounds of the type the model proposes .