PG-FMM: Physics-Guided Flow-Map Matching for Precipitation Nowcasting

Abstract

What a nowcast must get right

Two things — where the storm goes and what its fine structure looks like. Optimizing them together produces conditional-mean blur; optimizing structure alone drifts off the physical track.

PG-FMM separates the two. A frozen Lagrangian advection prior supplies an interpretable, motion-consistent forecast, and a few-step flow-map generator renders the unpredictable detail on top — conditioning on the prior rather than summing it, so neither module contaminates the other. Across four radar benchmarks (SEVIR, MeteoNet, CIKM, Shanghai) PG-FMM improves over the strongest published baseline on 18 of 24 metrics, with the largest gains on the heavy-rain thresholds where mean-seeking models fail — and it does so in only four network evaluations.

Method

Two stages, cleanly decoupled

PG-FMM architecture: Stage 1 Lagrangian prior (U-Net motion + source, semi-Lagrangian rollout); Stage 2 Flow-Map head conditioned on the concatenation of past frames and prior rollout, with a K=16 PMM ensemble. — PG-FMM = a frozen Lagrangian advection prior (Stage 1) conditioning a Flow-Map Matching head (Stage 2). Inference draws a \(K=16\) ensemble at \(\mathrm{NFE}=4\) and combines it with the Probability-Matched Mean.

1. Lagrangian prior (physics, frozen)

The predictable part of the field is transport. The prior enforces the advection equation

\[ \partial_t R + (v \cdot \nabla)\, R = s, \]

where a U-Net predicts the motion field \(v_t\) and source/sink \(s_t\), and a semi-Lagrangian scheme rolls the field forward: \( R_{t+1} = \operatorname{warp}(R_t, v_t) + s_t \).

2. Flow-Map Matching (generative)

Rather than integrating an instantaneous velocity field step by step, the head learns the two-time solution operator \( \Phi_\theta \) of the generative flow along the interpolant \( x_s = (1-s)\,x_0 + s\,x_1 \):

\[ \mathcal{L}_{\mathrm{FMM}} = \mathbb{E}_{\,0 \le r < t \le 1}\, \bigl\lVert\, \Phi_\theta\!\left(x_t,\, t \!\to\! r \mid c\right) - x_r \,\bigr\rVert_2^2, \qquad c = \bigl[\, R_{1:T_{\mathrm{in}}} \,;\; R^{\mathrm{prior}} \,\bigr]. \]

One network then serves any step budget — a single large shortcut or a few refinements — since valid maps compose: \( \Phi_{t \to r} = \Phi_{m \to r} \circ \Phi_{t \to m} \).

Quantitative

Comparison on four radar benchmarks

AlphaPre evaluation protocol (shared evaluator, thresholds, and test splits); our model uses a 16-member probability-matched-mean ensemble. Bold = best, underline = second best. ND = no explicit dynamics module, D = with one. Our method wins all six metrics on SEVIR and MeteoNet, four of six on Shanghai, and two of six on CIKM (18 / 24 overall).

SEVIR

Model	Type	CSI-M ↑	CSI-181 ↑	CSI-219 ↑	HSS ↑	SSIM ↑	MSE ↓
ConvGRU	ND	0.2903	0.0879	0.0350	0.3619	0.6100	368.34
MAU	ND	0.3076	0.1071	0.0516	0.3863	0.6505	355.48
SimVP	ND	0.3108	0.1106	0.0517	0.3924	0.6508	383.56
FourCastNet	ND	0.2686	0.0717	0.0339	0.3355	0.5976	410.27
Earthformer	ND	0.2892	0.0844	0.0245	0.3665	0.6633	360.11
PhyDNet	D	0.3017	0.1040	0.0278	0.3812	0.6532	357.63
Earthfarsser	D	0.3004	0.0992	0.0413	0.3829	0.6327	388.91
NowcastNet	D	0.2791	0.0770	0.0351	0.3512	0.6839	412.94
DiffCast	D	0.3050	0.1300	0.0582	0.3996	0.6482	559.59
AlphaPre	D	0.3259	0.1332	0.0545	0.4110	0.6884	345.18
Ours	D	0.3614	0.1859	0.0925	0.4593	0.7291	315.67

MeteoNet

Model	Type	CSI-M ↑	CSI-24 ↑	CSI-32 ↑	HSS ↑	SSIM ↑	MSE ↓
ConvGRU	ND	0.3401	0.2990	0.1431	0.4667	0.7833	12.85
MAU	ND	0.3233	0.2839	0.0997	0.4452	0.7897	12.92
SimVP	ND	0.3351	0.3002	0.1130	0.4573	0.7804	13.45
FourCastNet	ND	0.3027	0.2533	0.1085	0.4216	0.6450	15.05
Earthformer	ND	0.3205	0.2884	0.1237	0.4491	0.7772	14.43
PhyDNet	D	0.3384	0.3194	0.1366	0.4673	0.7823	14.48
Earthfarsser	D	0.3404	0.3170	0.1372	0.4726	0.7542	14.10
NowcastNet	D	0.3427	0.3206	0.1598	0.4751	0.7879	15.64
DiffCast	D	0.3512	0.3340	0.1808	0.4846	0.7887	17.93
AlphaPre	D	0.3824	0.3633	0.2002	0.5164	0.7968	12.74
Ours	D	0.4239	0.4066	0.2293	0.5583	0.8426	9.25

Shanghai

Model	Type	CSI-M ↑	CSI-35 ↑	CSI-40 ↑	HSS ↑	SSIM ↑	MSE ↓
ConvGRU	ND	0.3612	0.3163	0.2062	0.4899	0.7796	33.56
MAU	ND	0.3983	0.3621	0.2417	0.5346	0.7195	30.40
SimVP	ND	0.3850	0.3549	0.2382	0.5194	0.7795	34.40
FourCastNet	ND	0.3571	0.3108	0.2073	0.4868	0.5598	32.10
Earthformer	ND	0.3503	0.3178	0.1872	0.4844	0.7298	35.57
PhyDNet	D	0.3654	0.3236	0.2176	0.4957	0.7751	36.41
Earthfarsser	D	0.3926	0.3608	0.2343	0.5330	0.5405	32.68
NowcastNet	D	0.3953	0.3608	0.2450	0.5334	0.7902	33.56
DiffCast	D	0.4089	0.3740	0.2606	0.5476	0.7879	36.35
AlphaPre	D	0.4178	0.3854	0.2615	0.5534	0.7951	28.02
Ours	D	0.4181	0.3819	0.2636	0.5520	0.7962	26.05

CIKM

Model	Type	CSI-M ↑	CSI-35 ↑	CSI-40 ↑	HSS ↑	SSIM ↑	MSE ↓
ConvGRU	ND	0.3091	0.2009	0.1259	0.4006	0.6507	37.13
MAU	ND	0.3039	0.2054	0.1241	0.3928	0.6325	40.74
SimVP	ND	0.3052	0.2044	0.1321	0.3955	0.6538	38.06
FourCastNet	ND	0.2980	0.1849	0.1015	0.3801	0.4359	36.14
Earthformer	ND	0.3077	0.2039	0.1369	0.4001	0.6267	36.49
PhyDNet	D	0.3038	0.2052	0.1287	0.3931	0.6541	39.56
Earthfarsser	D	0.3000	0.2046	0.1259	0.3911	0.6373	39.87
NowcastNet	D	0.2991	0.1940	0.1188	0.3865	0.6713	40.96
DiffCast	D	0.3159	0.2009	0.1457	0.4085	0.6499	42.78
AlphaPre	D	0.3194	0.2068	0.1416	0.4137	0.6568	35.18
Ours	D	0.3149	0.2218	0.1540	0.4115	0.6252	42.18

The two highest (heavy-rain) thresholds are 181/219 (SEVIR), 24/32 (MeteoNet), and 35/40 (Shanghai and CIKM). \(K=16\) ensemble + Probability-Matched Mean at \(\mathrm{NFE}=4\).

Physics-Guided Flow-Map Matching for Precipitation Nowcasting