Harmonic Path Integral Diffusion

Harmonic Path Integral Diffusion (H-PID) introduces a novel approach to sampling from complex, continuous probability distributions by creating a time-dependent ‘‘bridge’’ from an initial point to the target distribution. Formulated as a Stochastic Optimal Control problem, H-PID balances control effort and accuracy through a unique three-level integrable structure: Top Level: Potential, force, and gauge terms combine to form a linearly solvable Path Integral Control system based on Green functions. Mid Level: With quadratic potentials and affine force/gauge terms, the Green functions reduce to Gaussian forms, mirroring quantum harmonic oscillators in imaginary time. Bottom Level: For a uniform quadratic case, the optimal drift/control reduces to a convolution of the target distribution with a Gaussian kernel, enabling efficient sampling. Implementation-wise the low-level H-PID operates without neural networks, allowing it to run efficiently on standard CPUs while achieving high precision. Validated on Gaussian mixtures and CIFAR-10 images, H-PID reveals a ‘‘weighted state’’ parameter as an order parameter in a dynamic phase transition, signaling early completion of the sampling process. This feature positions H-PID as a strong alternative to traditional methods sampling, such as simulated annealing, particularly for applications that demand analytical control, computational efficiency, and scalability.