# Differentiable Simulation

A finite-element solver for conjugate heat transfer and buoyant flow is recast as fixed-weight graph programs inside an automatic-differentiation framework, with assembled operators and the algebraic-multigrid cycle executed as sparse graph operations and nothing trained. With discretization, mesh, time step and tolerances held fixed, the same problem runs up to 181 times faster per step on one H200 than on 32 CPU cores, and the pressure solve alone 18.7 times faster. Reverse-mode gradients match central differences, and implicit differentiation holds one gradient near 4.9 forward solves.

A finite-element solver for conjugate heat transfer and buoyant flow is recast as fixed-weight graph programs inside an automatic-differentiation framework, with assembled operators and the algebraic-multigrid cycle executed as sparse graph operations and nothing trained. With discretization, mesh, time step and tolerances held fixed, the same problem runs up to 181 times faster per step on one H200 than on 32 CPU cores, and the pressure solve alone 18.7 times faster. Reverse-mode gradients match central differences, and implicit differentiation holds one gradient near 4.9 forward solves.

Surgical robots need to anticipate how instrument motion will deform tissue when visibility is interrupted, and prior differentiable physics simulation predicted absolute tissue positions, so reconstruction error, initialization, and simulation offsets left the whole surface persistently displaced. The forecast now anchors each tissue point to its last valid observation and predicts only the deformation induced by the known future tool trajectory, using mesh-free position-based dynamics with scene-specific rigidity, attachment, and grasp coupling. Across 12 surgical manipulation clips, the referenced forecast has a median within-clip error ratio of 0.912 against the persistence baseline, 95% interval 0.787–1.070, improving 8 of 12 clips, with the interval including parity.

Surgical robots need to anticipate how instrument motion will deform tissue when visibility is interrupted, and prior differentiable physics simulation predicted absolute tissue positions, so reconstruction error, initialization, and simulation offsets left the whole surface persistently displaced. The forecast now anchors each tissue point to its last valid observation and predicts only the deformation induced by the known future tool trajectory, using mesh-free position-based dynamics with scene-specific rigidity, attachment, and grasp coupling. Across 12 surgical manipulation clips, the referenced forecast has a median within-clip error ratio of 0.912 against the persistence baseline, 95% interval 0.787–1.070, improving 8 of 12 clips, with the interval including parity.