Manifold-Projected Diffusion Model for Generating Physical Visual Fields

Diffusion models are expressive generative priors for physical fields, but standard formulations typically treat coupled field variables as free outputs even when some are deterministically determined by known physical operators. As a result, generated parameter–state pairs can be individually plausible yet mutually inconsistent under the governing physics, particularly when forcing or boundary conditions shift beyond those seen during training. We propose the Manifold-Projected Diffusion Model (MPDM), which factorizes physical-field generation according to this dependency structure. Instead of directly denoising the complete field stack, MPDM predicts a compact latent representation of the independent physical parameter field, decodes it into a feasible parameter map, and uses a differentiable numerical solver to render the dependent state fields under prescribed conditions. This construction restricts each clean-state prediction to a solver-rendered, operator-consistent manifold while allowing reconstruction losses to propagate through the solver for end-to-end training without an auxiliary physics-residual objective. Across multiple PDE systems under controlled distribution shifts, MPDM consistently reduces state reconstruction error relative to ambient full-state and residual-based diffusion, while achieving strong parameter recovery and inverse consistency. These results demonstrate the benefit of aligning generative parameterization with the dependency structure imposed by the governing physics.