Source Anchoring for Physical Consistency in Flow Matching Models

Giulia Romoli; Filippo Ruffini; Paolo Soda

cs.LGSep 27, 2026

Source Anchoring for Physical Consistency in Flow Matching Models

Authors: Giulia Romoli, Filippo Ruffini, Paolo Soda

Organizations: Department of Diagnostics and Intervention, Ume˚a University, Sweden · Unit of Artificial Intelligence and Computer Systems, Universit`a Campus Bio-Medico di Roma, Italy

Abstract

Deep generative models are used to solve partial differential equations and model distributions of physical system states, but ensuring that the generated samples satisfy the governing laws remains challenging. Projection-based flow-matching methods enforce physics by correcting the flow from an unconstrained noise distribution. These corrections shift the generated samples away from the distribution of target solutions, especially in high noise regions. To address this limitation, we propose Source Anchoring for Physical Consistency (SAPC), a Functional Flow Matching method that encodes the physical constraints into the source noise before generation begins. We evaluate SAPC on five systems governed by partial differential equations, covering six tasks with linear and non-linear dynamics, and compare results against five baselines and the unconstrained backbone. Anchoring the source reduces the need for large corrections that drive samples onto admissible but off-distribution states, and SAPC reproduces the target distributions most accurately on every evaluated task, while matching the constraint precision of the best projection-based baselines. Ablation experiments show that this gain arises from pairing source projection with a matched training objective that regresses toward the projected source. These results identify the source distribution as a key design choice for physically consistent generative modelling.

Figures & tables

Figure 1: Left: existing projection-based flow-matching models correct the diffusion flow onto the manifold of admissible states to enforce physics, and the generated distribution (orange) may drift from the target one (green). Right: SAPC projects the noise on the manifold, anchoring it before diffusion starts (blue), mitigating the need for large corrections and reducing the distributional drift.

Figure 2: SAPC training (left) and sampling (right). SAPC anchors the source onto the physically-consistent manifold. The posterior-mean estimate is additionally projected at every Euler step.

Figure 3

Dataset	Dimensionality	Domain	Resolution	Local	Global
Heat	1D	$[0,2\pi]\times[0,1]$	$100\times 100$	IC	LMC
RD	1D	$[0,1)\times[0,1]$	$128\times 100$	IC	NLMC
Stokes	1D	$[0,1]\times[0,1]$	$100\times 100$	IC	NLEC
Stokes	1D	$[0,1]\times[0,1]$	$100\times 100$	BC	NLEC
Burgers	1D	$[0,1)\times[0,1]$	$100\times 100$	IC	LMC
NS	2D	$[0,1]^{2}\times[0,49]$	$[64,64]\times 50$	init. vorticity	LVC

Table 1: Datasets specifications: dimensionality; spatial and temporal domain; spatial and temporal resolution (number of discretization points); local constraint and global conservation law. For the local constraint, either IC/BC or a distributional parameter is held out (like initial vorticity for Navier-Stokes). For the global conservation law either LMC/NLMC, NLEC, or LVC.

	Metric	SAPC	PCFM	ECI	CAFM	PBFM	D-Flow	FFM
Heat	MMSE $\downarrow$	$\bm{2.2{\scriptstyle(0.8){-05}}}$	$\underline{1.2{\scriptstyle(0.2){-02}}}$	${1.9{\scriptstyle(0.6){-02}}}$	${2.2{\scriptstyle(0.6){-02}}}$	${5.8{\scriptstyle(1.3){-02}}}$	${8.3{\scriptstyle(6.0){-02}}}$	${5.7{\scriptstyle(1.3){-02}}}$
	SMSE $\downarrow$	$\bm{2.3{\scriptstyle(0.4){-05}}}$	${6.4{\scriptstyle(0.5){-02}}}$	${7.7{\scriptstyle(3.0){-02}}}$	${1.2{\scriptstyle(0.3){-01}}}$	${3.8{\scriptstyle(0.1){-02}}}$	${1.5{\scriptstyle(1.2){+01}}}$	$\underline{3.7{\scriptstyle(0.1){-02}}}$
	FPD $\downarrow$	$\bm{4.6{\scriptstyle(1.3){-03}}}$	${5.7{\scriptstyle(0.9){+00}}}$	${7.7{\scriptstyle(2.4){+00}}}$	${1.1{\scriptstyle(0.3){+01}}}$	${2.8{\scriptstyle(0.3){+00}}}$	${1.5{\scriptstyle(0.2){+01}}}$	$\underline{2.7{\scriptstyle(0.3){+00}}}$
	$\mathrm{CE}$ $\downarrow$	$\bm{2.0{\scriptstyle(0.0){-06}}}$	$\underline{2.0{\scriptstyle(0.0){-06}}}$	${2.6{\scriptstyle(0.1){-06}}}$	${2.2{\scriptstyle(0.1){-06}}}$	${6.5{\scriptstyle(0.3){-01}}}$	${2.2{\scriptstyle(0.9){+00}}}$	${1}$
RD	MMSE $\downarrow$	$\bm{5.8{\scriptstyle(0.9){-05}}}$	${1.9{\scriptstyle(0.6){-02}}}$	$\underline{5.8{\scriptstyle(1.6){-03}}}$	${2.3{\scriptstyle(0.7){-02}}}$	${4.1{\scriptstyle(1.1){-02}}}$	${5.6{\scriptstyle(4.0){-02}}}$	${4.2{\scriptstyle(1.2){-02}}}$
	SMSE $\downarrow$	$\bm{1.9{\scriptstyle(0.2){-05}}}$	${9.4{\scriptstyle(1.5){-02}}}$	${3.4{\scriptstyle(0.2){-02}}}$	${5.5{\scriptstyle(0.1){-02}}}$	${3.4{\scriptstyle(0.0){-02}}}$	${4.8{\scriptstyle(4.4){+01}}}$	$\underline{3.2{\scriptstyle(0.1){-02}}}$

Table 2: Generation on PDEs with linear and non-linear dynamics. Each cell reports the mean

m

and standard error

s

across seeds and held-out test sets in the compact form

m(s)p

, meaning

(m\pm s)\times 10^{p}

. For example,

2.2(0.8)-5

denotes

(2.2\pm 0.8)\times 10^{-5}

. Lower values indicate better performance, with best result in bold and second best underlined. Cell shading ranges from dark green (lowest mean) to white (highest mean) on a logarithmic scale, with normalization applied within each row.

Figure 3: Pointwise mean (top) and standard deviation (bottom) over generated test RD trajectories.

Figure 4: Wasserstein distance $W_{2}$ against the constraint error $\mathrm{CE}$ (both on log scales), with marker colour encoding wall-clock inference time per sample. Arrows point to the bottom-left corner, where both metrics are jointly minimised. Squares are SAPC and its ablated variants (S1, S2, and S3).

Figure 5: Trajectory analysis (non-linear RD, top, and linear Burgers, bottom). Left: residual of endpoint estimate, $\|\mathcal{R}(\mathcal{V}_{0,\theta})\|_{2}$ , before projection. Middle: residual $\|\mathcal{R}(\mathcal{V}_{t})\|_{2}$ till last reverse step. Right: deviation $\Delta$ of the trajectory from the FM interpolant; the inset bars report $\mathrm{KE}$ .

Appendix figures & tables16 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 6: Representative test-set trajectories of the heat benchmark, one per held-out IC. Each panel is drawn from a different test split, pinned to a distinct initial phase

\phi\in\{\pi/4,\pi/2,3\pi/4,\pi\}

, while the diffusivity

\alpha

is drawn from the training range

[1,5]

within each split. The heat solution

u(x,t)=e^{-\alpha t}\sin(x+\phi)

decays exponentially in time while preserving the spatial profile set by

\phi

Figure 7: Representative test-set trajectories of the RD benchmark, one per held-out IC. Each panel is drawn from a different test split, pinned to a distinct random low-mode Fourier draw $u_{\text{IC}}(x,0)$ , while the Neumann boundary fluxes $(g_{L},g_{R})$ vary per sample, drawn from the same training ranges within each split. The solution’s dynamics are dominated by diffusive smoothing of the initial fine-scale Fourier structure and by mass injected or removed through the Neumann boundary fluxes.

Panel of Figure 8 — Figure 8: Representative test-set samples for the two Stokes tasks. Both solve the classic Stokes velocity field $u(x,t)$ , a wave that decays exponentially away from the oscillating boundary at $x=0$ , with decay rate and spatial wave number both set by $k$ and $\omega$ . IC task (top): One sample from each of the four held-out test sets, each pinning a distinct decay rate $k\in\{4,8,12,16\}$ , while $\omega$ varies per sample, drawn from the training range $[2,8]$ . Larger $k$ gives a thinner boundary layer and faster spatial oscillation near $x=0$ . BC task (bottom): One sample from each of the four held-out test sets, each pinning a distinct frequency $\omega\in\{3,4.5,6,7.5\}$ , while $k$ varies within its training range $[2,20]$ per sample. Larger $\omega$ completes more oscillation periods over $t\in[0,1]$ .

Figure 9: Representative test-set trajectories of the Burgers benchmark. Each panel is drawn from a different test split, pinned to a distinct random low-mode Fourier draw

u_{\text{IC}}(x,0)

, while the viscosity

\nu

varies per sample, drawn from the training range

[0.01,0.05]

. Non-linear advection steepens the initial profile within the first few time steps, producing sharp layers whose thickness is set by

\nu

; the field then relaxes into near-uniform plateaus separated by these thin layers from then on.

Figure 10: One representative test-set trajectory of the 2D Navier-Stokes benchmark for each of its four held-out IC, shown as a $4\times 4$ grid: each row is a different test split pinned to a distinct unseen initial vorticity field $\omega_{0}$ drawn from a Gaussian random field; each column is one of four evenly spaced snapshots across the rollout $t\in[0,49]$ . The small-scale initial vorticity is rapidly overtaken by the large-scale pattern imposed by the diagonal forcing mode, with $\phi$ varying across trajectories.

	2D FNO	3D FNO
Fourier layers	4	4
Fourier modes per axis	32	16
Hidden channels	64	32
Projection channels	256	128
Time-embedding channels	32	16
Training batch size	256	32

Appendix

Table 3: FNO backbone configurations. 2D FNO for 1D datasets; 3D FNO for 2D datasets.

Benchmark	Projector	Max iters.	Tolerance	Damping $\mu$	Final $t=0$ pass
Heat	affine (exact)	1	-	-	no
Burgers	affine (exact)	1	-	-	no
NS	affine (exact)	1	-	-	no
RD	Gauss-Newton	3	$10^{-6}$	$10^{-8}$	yes
Stokes IC / BC	Gauss-Newton	10	$10^{-6}$	$10^{-8}$	yes

Appendix

Table 4: Sampling-time projector configuration. Affine families admit an exact one-shot projection; the curved ones use damped Gauss-Newton with an early stop on

\lVert h\rVert_{\infty}

and a final projection at

t=0

	Metric	SAPC	PCFM	ECI	CAFM	PBFM	D-Flow	FFM
	NFE	$100$	$100$	$500$	$100$	$100$	$2100$	$100$
Heat	IT [ms]	$100$	$43.2$	$497$	$44.0$	$\bm{20.8}$	$2739$	$20.7$
	$\times$ FFM	$4.85$	$2.09$	$24.02$	$2.13$	$1.00$	$133$
	T ${}_{\text{tot}}$ [s]	$123$	$52.9$	$608$	$53.9$	$\bm{25.4}$	$3356$	$25.3$
RD	IT [ms]	$127$	$78.6$	$862$	$80.9$	$\bm{25.5}$	$2755$	$26.0$
	$\times$ FFM	$4.91$	$3.03$	$33.21$	$3.12$	$0.98$	$106$

Appendix

Table 5: Inference cost of each sampler, measured on one NVIDIA GH200 96GB GPU. NFE is the number of network forward evaluations per sample; IT is the wall-clock time per generated sample,

\times

FFM that time relative to the unconstrained FFM baseline, and T

{}_{\text{tot}}

the wall-clock of the complete evaluation run (

n=1225

samples,

n=100

on Navier-Stokes). Each cell is the median over 3 seeds

\times

4 held-out test sets.

		SAPC	SAPC (S3)	SAPC (S2)	SAPC (S1)	FFM	condFFM
	training	$\checkmark$	$\times$	$\checkmark$	$\times$	$\times$	$\times$
	conditioning	$\times$	$\times$	$\times$	$\times$	$\times$	$\checkmark$
	$P(\mathcal{V}_{1})$	$\checkmark$	$\checkmark$	$\times$	$\times$	$\times$	$\times$
	$P(\mathcal{V}_{0,\theta})$	$\checkmark$	$\times$	$\checkmark$	$\checkmark$	$\times$	$\times$
Heat	MMSE	$\bm{2.2{\scriptstyle(0.8){-}05}}$	$1.5{\scriptstyle(0.5){-}02}$	$3.9{\scriptstyle(0.9){-}02}$	$4.0{\scriptstyle(1.0){-}02}$	$5.7{\scriptstyle(1.3){-}02}$	$2.1{\scriptstyle(0.8){-}04}$
	SMSE	$\bm{2.3{\scriptstyle(0.4){-}05}}$	$2.1{\scriptstyle(0.0){-}02}$	$2.3{\scriptstyle(0.0){-}02}$	$2.5{\scriptstyle(0.0){-}02}$	$3.7{\scriptstyle(0.1){-}02}$	$1.4{\scriptstyle(0.5){-}05}$

Appendix

Table 6: Ablation of SAPC over the constraint-aware training, the projection

P(\mathcal{V}_{1})

of the source, and the projection

P(\mathcal{V}_{0,\theta})

of the endpoint estimate at every reverse step. FFM and condFFM are also reported. Each cell reports the mean

m

and standard error

s

across seeds and held-out test sets in the compact form

m(s)p

, meaning

(m\pm s)\times 10^{p}

. Lower values indicate better performance, with the best result in bold and the second best underlined.

	Metric	SAPC	PCFM	ECI	CAFM	PBFM	D-Flow	FFM
Heat	W 2	$\bm{1.4{\scriptstyle(0.1){-}02}}$	$2.6{\scriptstyle(0.1){-}01}$	$2.6{\scriptstyle(0.5){-}01}$	$3.3{\scriptstyle(0.5){-}01}$	$2.3{\scriptstyle(0.2){-}01}$	$2.3{\scriptstyle(1.0){+}00}$	$2.3{\scriptstyle(0.1){-}01}$
	$\mathrm{CE}$ L	$6.0{\scriptstyle(2.3){-}11}$	$\bm{7.7{\scriptstyle(4.2){-}14}}$	$6.2{\scriptstyle(0.1){-}08}$	$3.9{\scriptstyle(2.1){-}13}$	$6.6{\scriptstyle(0.5){+}00}$	$3.1{\scriptstyle(0.8){+}00}$	$6.6{\scriptstyle(0.5){+}00}$
	$\mathrm{CE}$ G	$\bm{5.9{\scriptstyle(0.0){-}06}}$	$6.0{\scriptstyle(0.1){-}06}$	$7.9{\scriptstyle(0.4){-}06}$	$6.6{\scriptstyle(0.4){-}06}$	$4.3{\scriptstyle(0.0){-}01}$	$5.8{\scriptstyle(2.7){+}00}$	$1.5{\scriptstyle(0.1){+}00}$
RD	W 2	$\bm{7.0{\scriptstyle(0.2){-}03}}$	$3.2{\scriptstyle(0.2){-}01}$	$1.9{\scriptstyle(0.1){-}01}$	$2.7{\scriptstyle(0.1){-}01}$	$2.5{\scriptstyle(0.2){-}01}$	$3.2{\scriptstyle(1.8){+}00}$	$2.5{\scriptstyle(0.2){-}01}$
	$\mathrm{CE}$ L	$\bm{1.9{\scriptstyle(0.2){-}14}}$	$1.1{\scriptstyle(0.1){-}12}$	$1.4{\scriptstyle(0.0){-}07}$	$2.2{\scriptstyle(0.2){-}13}$	$4.9{\scriptstyle(0.1){+}00}$	$5.6{\scriptstyle(2.0){+}00}$	$4.9{\scriptstyle(0.1){+}00}$
	$\mathrm{CE}$ G	$\bm{3.1{\scriptstyle(0.2){-}07}}$	$3.1{\scriptstyle(0.2){-}07}$	$3.4{\scriptstyle(0.1){-}07}$	$3.1{\scriptstyle(0.1){-}07}$	$1.7{\scriptstyle(0.2){-}02}$	$2.8{\scriptstyle(2.5){+}00}$	$4.7{\scriptstyle(0.0){-}02}$

Appendix

Table 7: Supplement to Table 2 : the Wasserstein-2 distance (

W_{2}

) and the two constraint-errors

\mathrm{CE}

L and

\mathrm{CE}

G (see Section C.3 for their definition). Each cell reports the mean

m

and standard error

s

across seeds and held-out test sets in the compact form

m(s)p

, meaning

(m\pm s)\times 10^{p}

. Lower values indicate better performance, with the best result in bold and the second best underlined

Figure 11: Generated trajectories on Heat for one test split: pointwise mean (top) and standard deviation (bottom) over $1225$ trajectories sharing the split’s IC at $\phi=\pi/4$ .

Figure 12: Generated trajectories on Stokes for one test split: pointwise mean (top) and standard deviation (bottom) over $1225$ trajectories sharing the split’s IC at $k=4$ , differing in forcing frequency.

Figure 13: Generated trajectories on Stokes for one test split: pointwise mean (top) and standard deviation (bottom) over $1225$ trajectories sharing the split’s BC at $\omega=3$ , differing in decay rate.

Figure 14: Generated trajectories on Burgers for one test split: pointwise mean (top) and standard deviation (bottom) over $1225$ trajectories sharing the split’s IC.

Figure 15: Generated trajectories on Navier-Stokes for one test split. Each column tiles $10$ vorticity frames left to right in $t$ ; trajectories share the initial vorticity and differ in the forcing phase.

Figure 16: The same as Figure 5 , but for the Heat, Stokes (IC and BC) and Navier-Stokes datasets.

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Constraint-Aware Flow Matching: Decision Aligned End-to-End Training for Constrained Sampling

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University of Virginia · NASA Langley Research Center

Generative Models Contrastive Flow Matching Training-Inference Mismatch Decisions

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Deep generative models provide state-of-the-art performance across a wide array of applications, with recent studies showing increasing applicability for science and engineering. Despite a growing corpus of literature focused on the integration of physics-based constraints into the generation process, existing approaches fail to enforce strict constraint satisfaction while maintaining sample quality. In particular, training-free constrained sampling methods, while providing per-sample feasibility guarantees, introduce a fundamental mismatch between the training objective and the constrained sampling procedure, often leading to performance degradation. Identifying this training-sampling misalignment as a central limitation of current constrained generative modeling approaches, this paper proposes Constraint-Aware Flow Matching, a novel end-to-end framework that explicitly incorporates constraint projections into the training objective. By aligning the model's learned dynamics with the constrained sampling process, the proposed method mitigates distributional shift induced by projection-based corrections, enabling high-quality constrained generation. The proposed approach is evaluated on three challenging real-world benchmarks, illustrating the generality and efficacy of the method.

Abstract

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