  Thermodynamic integration (TI) offers a rigorous method for estimating
free-energy differences by integrating over a sequence of interpolating
conformational ensembles. However, TI calculations are computationally
expensive and typically limited to coupling a small number of degrees of
freedom due to the need to sample numerous intermediate ensembles with
sufficient conformational-space overlap. In this work, we propose to perform TI
along an alchemical pathway represented by a trainable neural network, which we
term Neural TI. Critically, we parametrize a time-dependent Hamiltonian
interpolating between the interacting and non-interacting systems, and optimize
its gradient using a denoising-diffusion objective. The ability of the
resulting energy-based diffusion model to sample all intermediate ensembles
allows us to perform TI from a single reference calculation. We apply our
method to Lennard-Jones fluids, where we report accurate calculations of the
excess chemical potential, demonstrating that Neural TI is capable of coupling
hundreds of degrees of freedom at once.
