Multiphase-Diff: Diffusion-Based Generative Modeling for High-Contrast Multiphase Physical Systems with Sharp Interfaces

Physics-informed generative models augment a data-fitting objective with residual penalties that encourage physical constraints to hold in generated samples. However, improving constraint satisfaction need not improve fidelity to the data distribution. We study this conflict in residual-regularized diffusion models and identify two possible consequences. First, the same prediction error can incur different residual costs near different feasible states, preferring states with lower residual sensitivity (avoidance). Second, this preference can extend to feasible states outside the data support, where the residual also vanishes (leakage). Motivated by this analysis, we propose a data-anchored penalty that weights prediction errors by the residual Jacobian at the clean training sample and normalizes the resulting magnitude. This removes the incentive to relocate to less sensitive states without changing the model architecture or sampling procedure. Experiments on a controlled synthetic system, three PDE systems, and molecular transition-state generation with flow matching demonstrate the benefits of anchoring. Code is available at https://github.com/TanjaBien/anchored_diffusion.

Diffusion models can be used to generate spatiotemporal signals of physical phenomena, such as time-series images of fluid dynamics. However, a major limitation of standard diffusion models is that they do not incorporate constraints derived from the underlying physical laws. Consequently, generated samples may appear visually plausible while deviating substantially from the true dynamics. In this study, we propose a simple yet effective physics-informed approach based on diffusion guidance with self-generated data augmentation. The proposed method learns the data distribution conditioned on the degree of deviation from the physically correct dynamics and generates samples by explicitly setting the deviation condition to be zero. The method decouples the evaluation of the governing equations from the diffusion model training and sampling processes, avoiding the need to solve the governing equations at every iteration of the denoising process. This design makes the method applicable to problems requiring computationally expensive numerical simulations and enables faster sample generation. Experimental results demonstrate that the proposed model not only significantly reduces the deviations compared with standard diffusion models but also achieves further reductions when combined with existing physics-constrained diffusion methods.

Probabilistic super-resolution of high-dimensional spatial fields using diffusion models is often computationally prohibitive due to the cost of operating directly in pixel space. We propose PODiff, a structured conditional generative framework that performs diffusion in a fixed, variance-ordered Proper Orthogonal Decomposition (POD) coefficient space, exploiting the orthogonality of POD modes to impose an interpretable, variance-ordered latent geometry. This design enables efficient ensemble generation, preserves dominant spatial structure, and yields spatially interpretable, well-calibrated uncertainty at substantially lower computational cost. We evaluate PODiff on sea surface temperature downscaling over the West Australian coast and on a controlled advection-diffusion benchmark. PODiff achieves reconstruction accuracy comparable to pixel-space diffusion while requiring significantly less memory and producing more reliable uncertainty estimates than deterministic and Monte Carlo Dropout baselines.