  In the field of generative modeling based on differential equations,
conventional methods utilize scalar-valued time during both the training and
inference phases. This work introduces, for the first time, a tensor-valued
time that expands the conventional scalar-valued time into multiple dimensions.
Additionally, we propose a novel path optimization problem designed to
adaptively determine multidimensional inference trajectories using a
predetermined differential equation solver and a fixed number of function
evaluations. Our approach leverages the stochastic interpolant framework,
simulation dynamics, and adversarial training to optimize the inference
pathway. Notably, incorporating tensor-valued time during training improves
some models' inference performance, even without path optimization. When the
adaptive, multidimensional path derived from our optimization process is
employed, further performance gains are achieved despite the fixed solver
configurations. The introduction of tensor-valued time not only enhances the
efficiency of models but also opens new avenues for exploration in training and
inference methodologies, highlighting the potential of adaptive
multidimensional paths.
