DUE: A Deep Learning Framework and Library for Modeling Unknown Equations

The paper defines OSG-Net as Fθ(u,Δ)=u+ΔNθ(u,Δ), which immediately implies Fθ(u,0)=u, i.e., Φ0=In, matching part (1) of the solver’s argument . The semigroup property ΦΔ1+Δ2=ΦΔ1∘ΦΔ2 is stated as the target structure (2.14b), and the paper enforces it via the GDSG semigroup-informed loss (3.3)–(3.4) that compares one-step (Δ0+Δ1) vs. two-step rollouts; if this residual vanishes for all u and positive Δ, the solver’s derivation correctly shows the semigroup law for all Δ≥0, consistent with the paper’s intent . For (3), the multi-step loss averages errors over K+1 future steps (3.2), directly penalizing rollout discrepancies and mitigating accumulation, aligning with the solver’s explanation . Minor note: the solver assumed both composition equalities (A) and (B); only one is needed to conclude the semigroup identity (order is a convention), but this redundancy does not affect correctness.