Discovery of differential equations using sparse state and parameter regression

The paper defines the mixed-regression losses Lλ and Lλ,R, introduces the smooth l0 surrogate ∥θ∥ε = Σi(1 − exp(−θi2/(2ε2))) and notes that limε→0 ∥·∥ε = ∥·∥0, then uses BIC for model selection and an acceptance rule that updates bestIC only upon strict improvement (eqs. (4), (5), (7), (9) and Algorithm 1) . The candidate solution supplies the missing mathematical details: a pointwise proof of the ε→0+ limit with monotonicity and an explicit exponential-in-1/ε2 error bound; explicit first-order gradients via the chain rule (with a selection matrix S formalizing u|D); and a formal statement that the recorded best BIC is nonincreasing and convergent. These are consistent with the paper’s setup and claims. One caveat: the candidate’s further claim that the sequence becomes constant after finitely many iterations requires an additional assumption (e.g., that each mask’s subproblem is solved to a unique global minimum and that accepted masks are never revisited). The paper does not make this claim, so there is no conflict; with this mild caveat, both are correct, with the model providing a more explicit proof of properties the paper states informally.