Non-Negative Universal Differential Equations With Applications in Systems Biology

The paper’s Theorem 1 is correct in substance, but its proof contains a gap: it asserts there exists a time t* with x_i(t*)=0 and (f_i+N_iU_i)(t*)<0 when a component becomes negative, which need not hold (a differentiable trajectory can cross zero with zero slope). The candidate solution supplies a rigorous alternative via a negative-part differential inequality and Grönwall, establishing forward invariance under the stated Lipschitz and boundary-vanishing assumptions for N. Hence, the result stands, but the paper’s proof sketch should be amended. See Theorem 1 statement and proof context in the PDF (model structure (3) and Theorem 1) and their proof by contradiction .