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

Yining Huang; Zhenyu Liang

cs.CVAug 13, 2026

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

Authors: Yining Huang, Zhenyu Liang

Organizations: University of Texas at Austin · The Hong Kong University of Science and Technology · University of Maryland, College Park

Abstract

Physics-constrained diffusion for high-contrast, sharp-interface multiphase fields faces three coupled difficulties. At coefficient jumps, expanded pointwise strong-form PDE residuals contain singular gradient terms that can penalize physical interfaces. Under extreme contrast, low-magnitude phases may fall below the diffusion noise floor and be erased, misscaled, or generated with negative coefficients, while a global likelihood scale allows high-magnitude phases to dominate supervision. We therefore propose Multiphase-Diff, which makes three corresponding contributions: (i) a conservative flux residual that avoids differentiating discontinuous coefficients and enforces discrete conservation; (ii) an analytic bijective representation that maps low-amplitude signals to order-one latent scales and guarantees coefficient positivity through exponential decoding; and (iii) a Jacobi-preconditioned likelihood that normalizes local residual scales for balanced supervision. Experiments on three complementary multiphase benchmarks demonstrate the superiority of Multiphase-Diff over seven baselines in both physical and distributional fidelity and its robustness across phase contrasts and compositions, establishing its effectiveness for scientific sample generation in this challenging regime.

Figures & tables

Figure 1: Overview of Multiphase-Diff, illustrated with a Darcy coefficient–solution pair. During training, the analytic bijection $g$ maps $(a,u)$ to the clean latent $z_{0}$ , which is corrupted by the forward process and reconstructed by a clean-latent-predicting U-Net. The prediction $\hat{z}_{0}$ is decoded to $(\hat{a},\hat{u})$ , whose conservative finite-volume flux residual is Jacobi-preconditioned to define the physics loss combined with the latent data loss. These mechanisms keep low-amplitude phases resolvable and coefficients positive, enforce interface-compatible local conservation, and balance residual scales across phases and samples, respectively. Inference requires no residual correction and performs standard reverse diffusion in latent space followed by one decoding step.

	DDPM	PG-Diffusion	CoCoGen	DiffusionPDE	FunDiff	PIDM	PIDM-log	Multiphase-Diff
(a) Physical Residual Fidelity (PRF) $\downarrow$
Case 1	$1.09\!\times\!10^{-2}$	$1.34\!\times\!10^{-2}$	$1.96\!\times\!10^{-2}$	$9.76\!\times\!10^{-3}$	$6.28\!\times\!10^{-3}$	$9.81\!\times\!10^{-4}$	$8.26\!\times\!10^{-4}$	$\boldsymbol{3.58\!\times\!10^{-4}}$
Case 2	$8.57\!\times\!10^{-2}$	$1.14\!\times\!10^{-1}$	$1.60\!\times\!10^{-1}$	$9.18\!\times\!10^{-2}$	$1.47\!\times\!10^{-3}$	$4.57\!\times\!10^{-1}$	$1.61\!\times\!10^{-3}$	$\boldsymbol{2.30\!\times\!10^{-4}}$
Case 3	$3.43\!\times\!10^{-3}$	$3.16\!\times\!10^{-3}$	$1.24\!\times\!10^{-3}$	$3.23\!\times\!10^{-3}$	$4.48\!\times\!10^{-3}$	$1.68\!\times\!10^{-3}$	$3.15\!\times\!10^{-3}$	$\boldsymbol{4.17\!\times\!10^{-4}}$
(b) Negative-coefficient rate (Neg, %) $\downarrow$
Case 1	1.9	4.5	6.9	2.8	0.0	2.4	0.0	0.0

Table 1: Baseline comparison over 256 generated samples. Lower is better for all metrics, and the best values are shown in bold.

Figure 2: Four-metric visual comparison across three benchmarks. PRF,

W_{1}

, and Sharp use log scales; Neg is linear. Shading highlights Multiphase-Diff’s superior performance.

Figure 3: Qualitative comparison of coefficient, solution, and Jacobi-preconditioned residual magnitude $|\widetilde{R}|$ across three cases. The strongest baseline is selected by the lowest PRF; green pixels in CoCoGen’s $\sigma$ field indicate non-positive conductivity.

Variant	PRF ( $\times 10^{-3}$ ) $\downarrow$	Worst $W_{1}$ $\downarrow$	Sharp $\downarrow$	Int.	PFE $\downarrow$
w/o Flux Res.	$0.826/1.61/3.15$	$0.179/1.422/1.282$	$0.055/0.128/0.193$	$62.2/0.2/164.3$	$0.098/0.080/0.013$
w/o Bijection	Diverged			$0/0/0$	$0.492/0.739/0.384$
w/o Jacobi	$\mathbf{0.085}/\mathbf{0.044}/1.05$	$2.021/2.647/0.982$	$0.137/0.166/0.213$	$0/0/2.4$	$0.316/0.098/0.032$
Multiphase-Diff	$0.358/0.230/\mathbf{0.417}$	$\mathbf{0.008}/\mathbf{0.933}/\mathbf{0.025}$	$\mathbf{0.023}/\mathbf{0.023}/\mathbf{0.039}$	$\mathbf{63.8}/\mathbf{64.2}/\mathbf{187.9}$	$\mathbf{0.045}/\mathbf{0.073}/\mathbf{0.010}$

Table 2: Component ablations on Cases 1/2/3. Lower is better for metrics marked by

\downarrow

; Int. diagnoses interface collapse and should remain comparable to that of the complete Multiphase-Diff model. Best values are bold.

Figure 4: Robustness study to phase contrast and composition. The upper and lower axes show the two sweeps; PRF remains of order

10^{-4}

and Neg is zero. The dashed line marks the reference-data floor.

Contrast sweep, $v_{f}=0.5$
Setting	PRF	Neg (%)	Worst $W_{1}$	Sharp
$\gamma=10^{1}$	$3.10{\times}10^{-4}$	0.0	0.009	0.006
$\gamma=10^{2}$	$3.50{\times}10^{-4}$	0.0	0.015	0.009
$\gamma=10^{3}$	$3.48{\times}10^{-4}$	0.0	0.013	0.009
$\gamma=10^{4}$	$4.07{\times}10^{-4}$	0.0	0.034	0.016
$\gamma=10^{5}$	$3.99{\times}10^{-4}$	0.0	0.029	0.021

Table 3: Complete results for the controlled contrast and composition sweeps reported in the main paper.

Method	Physics location	Representation	Residual	Weight	Training or sampling modification
DDPM	none	affine standardization	none	0	standard reverse diffusion
PG-Diffusion	conditioning	affine standardization	pointwise strong form	$10^{-3}$	two conditioning channels; guidance scale 3.0
CoCoGen	sampling correction	affine standardization	flux + Jacobi at sampling	0	$N=100$ , $M=10$ , $\eta=0.05$
DiffusionPDE	sampling guidance	affine standardization	flux + Jacobi at sampling	0	$\zeta=0.1$ over all $T$ steps
FunDiff	autoencoder prior	function autoencoder, 8 latent channels	physics-informed autoencoder	0.1	FAE 40k steps + rectified flow 60k steps
PIDM	training likelihood	affine standardization	pointwise strong form	$10^{-3}$	standard reverse diffusion

Table 4: Implementation settings for the seven baselines and Multiphase-Diff. Weights denote the training-time physics coefficient; inference-time methods reuse the physics-free DDPM checkpoint.

	Case 1		Case 2		Case 3
Method	low $K$	high $K$	gas	liquid	pore	active	binder
DDPM	0.144	0.006	1.141	0.538	2.522	0.682	0.001
PG-Diffusion	0.298	0.005	1.140	0.667	2.520	0.725	0.001
CoCoGen	0.116	0.005	1.149	0.580	2.715	0.508	0.001
DiffusionPDE	0.174	0.006	1.119	1.187	2.502	0.681	0.001
FunDiff	0.038	0.033	0.051	1.339	0.146	0.219	0.739

Table 5: Per-phase Wasserstein-1 distance of

\ln a

on the positive support. Lower is better. Non-positive coefficient values are excluded from this logarithmic distance and reported separately by Neg.

Case	Method	PRF median	PRF mean	PFE	Interfaces/sample	Neg (%)
1	DDPM	$1.09\times 10^{-2}$	$2.56\times 10^{-2}$	0.005	29.0	1.9
1	PG-Diffusion	$1.34\times 10^{-2}$	$3.98\times 10^{-2}$	0.011	40.7	4.5
1	CoCoGen	$1.96\times 10^{-2}$	$1.83\times 10^{-1}$	0.000	49.0	6.9
1	DiffusionPDE	$9.76\times 10^{-3}$	$4.77\times 10^{-2}$	0.052	29.9	2.8
1	FunDiff	$6.28\times 10^{-3}$	$7.26\times 10^{-3}$	0.025	93.1	0.0
1	PIDM	$9.81\times 10^{-4}$	$2.93\times 10^{-3}$	0.360	9.2	2.4

Table 6: Additional diagnostics for all eight methods. Lower is better for PRF, PFE, and Neg. Interface counts diagnose preservation or collapse and should be interpreted relative to the reference morphology.

Case	PRF	Worst $W_{1}$	Sharp
Darcy	$4.57\times 10^{-7}$	$0.0030\pm 0.0010$	$0.0069\pm 0.0031$
Gas–liquid	$3.69\times 10^{-7}$	$0.1406\pm 0.0799$	$0.0034\pm 0.0017$
Electrode	$2.17\times 10^{-7}$	$0.0000\pm 0.0000$	$0.0066\pm 0.0038$

Table 7: Reference-data floors under the exact evaluation conventions of the main paper.

Figure 5: Qualitative comparison of the reference data, all seven baselines, and Multiphase-Diff on Case 1. Columns correspond to methods; rows show permeability $K$ , pressure $p$ , and Jacobi-preconditioned conservative residual magnitude $|\widetilde{R}|$ ; green denotes non-positive permeability.

Figure 6: Qualitative comparison of the reference data, all seven baselines, and Multiphase-Diff on Case 2. Rows show viscosity $\mu$ , axial velocity $w$ , and Jacobi-preconditioned conservative residual magnitude $|\widetilde{R}|$ ; green denotes non-positive viscosity.

Figure 7: Qualitative comparison of the reference data, all seven baselines, and Multiphase-Diff on Case 3. Rows show conductivity $\sigma$ , electric potential $\varphi$ , and Jacobi-preconditioned conservative residual magnitude $|\widetilde{R}|$ ; green denotes non-positive conductivity.

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