On the order of Runge Kutta methods reusing last stage

What the math gives to ML

The paper gives an order-theoretic treatment of explicit Runge–Kutta schemes whose final derivative evaluation is reused as the first evaluation of the next step. This is directly transferable to weight-tied neural ODEs and continuous-depth residual networks, where a function evaluation means an expensive shared MLP, convolution, or transformer block. The engineering opportunity is to impose endpoint-identification constraints on the integrator so that one neural-vector-field call is removed from every step without changing the intended integration order.