tLaSDI: Thermodynamics-informed latent space dynamics identification

The paper’s Theorem 2 states the ROM error bound ∥e(t; t0)∥ ≲ ε_int + ε_rec + ε_Jac + ε_mod under bounded/Lipschitz J_d and bounded F^r, and proves it by introducing the ideal reconstruction x̂, splitting the error into e_ideal and e_ROM, and combining chain-rule-based bounds with Lipschitz estimates for J_d and boundedness of F^r . The candidate solution obtains the same bound by a near-identical mechanism—adding/subtracting the autoencoder reconstruction, differentiating G(s) = (φ_d∘φ_e)(x(s)) − φ_d(z(s)), and bounding the derivative using the same structural assumptions—then absorbing constants, exactly matching the paper’s error components and hidden constants . Minor stylistic differences (paper’s α-splitting versus the candidate’s min ≤ sum step) do not affect correctness.