Metriplectic Neural Source Layer

Integrate this at the residual-update or latent-dynamics stage of a neural simulator, rather than inside an ordinary unconstrained MLP output head. Represent each sample state as $x$ and provide differentiable scalar functions $H(x)$ and, if available, $S(x)$. Compute $g=\nabla_xH(x)$ with automatic differentiation. Define a fixed skew matrix $K$ or a sparse block operator matching the state variables, and calculate the conservative derivative $v_c=Kg$. Have an MLP $f_\theta(x)$ output one scalar per admissible transfer channel; convert these values to rates with $r=\operatorname{softplus}(f_\theta(x))$. Store fixed transfer vectors $s_j$ whose entries add and remove the appropriate conserved quantities. Compute $v_d=\sum_jr_js_j$ and use $\dot{x}=v_c+v_d$. If the model has chemical potentials, set each affinity to $a_j=s_j^\top\nabla S(x)$ and orient or gate channels so $a_j\ge0$; alternatively use $r_j=\operatorname{softplus}(f_j)\operatorname{softplus}(a_j)$ and include the signed entropy contribution explicitly. For a discrete residual block use $x_{k+1}=x_k+h(v_c+v_d)$, and compare it with a midpoint or symplectic integrator for the $Kg$ term. The paper supplies the algebraic conservation identity; estimate unknown $H$, $S$, or affinities from analytic simulator formulas or automatic differentiation, not a separately trained network. First test on a 2D damped Hamiltonian toy system and then a 1D reaction benchmark, using an unconstrained neural source baseline with the same parameter count. Track one-step error, cumulative relative energy drift $|H_T-H_0|/|H_0|$, entropy violations, and long-horizon rollout error. Success means comparable validation error with orders-of-magnitude smaller energy drift and fewer unstable rollouts at the same step size.