Physics-informed NeuralODE for Post-disaster Mobility Recovery (KDD2024 Research Track)

This repo contains codes and data for the following paper:

Jiahao Li, Huandong Wang, Xinlei Chen*: Physics-informed NeuralODE for Post-disaster Mobility Recovery

Environment building and code runing

The main environment requirements can be seen in requirements.txt

To run the training, testing and get the experimental result, just run

cd CDGON-KDD24
python main.py

About the data

The mobility data from Aug 1st - Sep 10th, 2019 in FL, GA and SC are collected in

./data

All data are organized into shapes like [T,N,N], representing T population mobility matrices, where diagonal elements represent intra-regional population flows and non-diagonal elements represent inter-regional population flows.

Supplementary materials, including proof of convergence of the key formula $\frac{\mathrm{d} r_i(t)}{\mathrm{d} t} = \alpha \frac{r_i(t)}{\overline{r_i}}[ \overline{r_i} - r_i(t)]$, performance evaluation results of STGCN and CDGON, and hyper-parameter experimental results:

Theorem: In the formula $\frac{\mathrm{d} r_i(t)}{\mathrm{d} t} = \alpha \frac{r_i(t)}{\overline{r_i}}[ \overline{r_i} - r_i(t)]$, $r_i(t)$ converges to $\overline{r_i}$ when $t \to \infty $ instead of oscillating perpetually around $\overline{r_i}$.

Proof of Theorem:

The differential equation $\frac{\mathrm{d} r_i(t)}{\mathrm{d} t} = \alpha \frac{r_i(t)}{\overline{r_i}}[ \overline{r_i} - r_i(t)]$ can be transformed to:

$\overline{r_i}\mathrm{d} r_i(t) = \alpha r_i(t)[ \overline{r_i} - r_i(t)] \mathrm{d} t$,

which is a separable variable equation and we can have:

$\frac{1}{\alpha r_i(t)[ \overline{r_i} - r_i(t)]}\mathrm{d} r_i(t) = \frac{1}{\overline{r_i}}\mathrm{d} t$,

which can be integrated to obtain:

$\frac{r_i(t)}{\overline{r_i} - r_i(t)} = Ce^{\alpha t}$,

which can generate:

$r_i(t) = \frac{\overline{r_i}}{1+Ce^{-\alpha t}} $,

which satisfies:

$\lim_{t \to \infty} r_i(t) = \lim_{t \to \infty}\frac{\overline{r_i}}{1+Ce^{-\alpha t}}=\overline{r_i}$

which proves the theorem

Comparision of Performance Evaluation between STGCN and CDGON

Experiments Type	Metrics	STGCN	CDGON
Performance Evaluation in FL	MAE	62417.8359	59767.4805
	R2	0.9909	0.9948
	NRMSE	0.0954	0.0724
Performance Evaluation in GA	MAE	7082.8677	2013.2821
	R2	0.9734	0.9977
	NRMSE	0.1631	0.0475
Performance Evaluation in in SC	MAE	18133.7500	9040.6785
	R2	0.9759	0.9941
	NRMSE	0.1554	0.0771

Hyper-parameter experimental results

Hyper-parameter settings in CDGON

Embedding dimension: 48

edge loss weight $\lambda$: 100

Learning rate: 0.003

Experiment results on different embedding dimensions, where the other parameter settings are same as original paper.

Experiments Type	Metrics	16	32	48	64	80
Performance Evaluation in FL	MAE	65796.6484	31556.1035	59767.4805	38900.8594	16794.2266
	R2	0.9817	0.9947	0.9948	0.9949	0.9993
	NRMSE	0.1353	0.0728	0.0724	0.0711	0.0265
Performance Evaluation in GA	MAE	8596.8066	2476.3357	2013.2821	6726.749	2004.8026
	R2	0.9	0.9965	0.9977	0.9649	0.9976
	NRMSE	0.3162	0.0588	0.0475	0.1873	0.0493
Performance Evaluation in in SC	MAE	15904.1963	20676.6699	9040.6758	13369.793	35273.7461
	R2	0.9814	0.9683	0.9941	0.9821	0.9332
	NRMSE	0.1363	0.178	0.0771	0.134	0.2584
Generalization FL -> GA	MAE	8991.2666	5983.022	5433.7192	3936.6899	12559.5605
	R2	0.9442	0.9704	0.9831	0.9852	0.8773
	NRMSE	0.2363	0.1722	0.1301	0.1215	0.3502
Generalization FL -> SC	MAE	22449.4219	15979.0742	13609.5889	13573.167	32977.6367
	R2	0.9525	0.9736	0.9773	0.9797	0.8973
	NRMSE	0.218	0.1625	0.1508	0.1426	0.3204
Generalization GA -> FL	MAE	44399.207	46987.5703	48276.1992	40045.4766	57892.6836
	R2	0.9901	0.9919	0.9901	0.9922	0.9863
	NRMSE	0.0997	0.0898	0.0997	0.0881	0.1171
Generalization GA -> SC	MAE	14188.7959	16384.1875	14315.9375	12567.7197	13666.5508
	R2	0.9721	0.9732	0.982	0.9847	0.9831
	NRMSE	0.167	0.1637	0.1341	0.1235	0.13
Generalization SC -> FL	MAE	37216.1172	42591.7539	73204.6719	53571.0898	74730.8906
	R2	0.9926	0.9916	0.9801	0.988	0.9782
	NRMSE	0.0859	0.0916	0.1412	0.1095	0.1478
Generalization SC -> GA	MAE	4730.644	3934.5408	8921.1035	3830.97	11805.2217
	R2	0.9785	0.9875	0.9687	0.9891	0.9456
	NRMSE	0.1467	0.1117	0.177	0.1042	0.2331

Experiment results on different $\lambda$, where the other parameter settings are same as original paper.

Experiments	Metrics	10	50	100	500	1000
Performance Evaluation in FL	MAE	32522.3301	40861.3164	59767.4805	35012.9688	64905.2461
	R2	0.994	0.9912	0.9948	0.9953	0.9866
	NRMSE	0.0772	0.0939	0.0724	0.0686	0.116
Performance Evaluation in GA	MAE	43049.1211	4385.7549	2013.2821	7148.8594	5631.6104
	R2	0.4618	0.9816	0.9977	0.9399	0.9876
	NRMSE	0.7336	0.1357	0.0475	0.2452	0.1113
Performance Evaluation in SC	MAE	21663.9062	22532.2617	9040.6758	16925.8652	38515.3086
	R2	0.9648	0.9658	0.9941	0.9674	0.8788
	NRMSE	0.1875	0.1848	0.0771	0.1804	0.3481
Generalization FL -> GA	MAE	7073.2612	7787.6982	5433.7192	4014.4309	8322.3545
	R2	0.9646	0.9603	0.9831	0.9835	0.9553
	NRMSE	0.1882	0.1993	0.1301	0.1283	0.2114
Generalization FL -> SC	MAE	17733.6484	18547.4883	13609.5889	14024.291	21716.9141
	R2	0.9692	0.9665	0.9773	0.9771	0.9559
	NRMSE	0.1756	0.1832	0.1508	0.1513	0.2099
Generalization GA -> FL	MAE	44205.918	45013.5977	48276.1992	50217.8906	52088.3398
	R2	0.9913	0.9912	0.9901	0.9889	0.9904
	NRMSE	0.0932	0.0937	0.0997	0.1053	0.0979
Generalization GA -> SC	MAE	12889.4541	13916.4736	14315.9375	13024.3271	12746.8926
	R2	0.9849	0.9823	0.982	0.986	0.986
	NRMSE	0.1227	0.1329	0.1341	0.1185	0.1184
Generalization SC -> FL	MAE	67278.0703	80161.0156	73204.6719	54324.0898	69783.5703
	R2	0.9834	0.9756	0.9801	0.9888	0.9717
	NRMSE	0.1288	0.1561	0.1412	0.1058	0.1681
Generalization SC -> GA	MAE	5476.9692	14294.9346	8921.1035	3708.6938	9658.623
	R2	0.9864	0.9228	0.9687	0.9922	0.9241
	NRMSE	0.1166	0.2779	0.177	0.0881	0.2756

Experiment results on different learning rates, where the other parameter settings are same as original paper.

Experiments	Metrics	0.0001	0.0005	0.001	0.005	0.01
Performance Evaluation in FL	MAE	529004.1875	34693.8633	33831.0703	36524.457	60011.1562
	R2	0.1695	0.9975	0.9976	0.9984	0.9942
	NRMSE	0.9113	0.0496	0.0488	0.0396	0.076
Performance Evaluation in GA	MAE	37850.4648	4462.7021	5688.2329	2161.2673	11902.7266
	R2	0.003	0.9876	0.9777	0.9972	0.9387
	NRMSE	0.9985	0.1112	0.1495	0.0526	0.2476
Performance Evaluation in SC	MAE	58461.9492	15060.959	18775.9277	32302.1719	19255.9629
	R2	0.7281	0.9858	0.9748	0.9269	0.9672
	NRMSE	0.5214	0.1194	0.1586	0.2703	0.1812
Generalization FL -> GA	MAE	46940.0625	39384.4102	6600.5337	4510.082	5471.5781
	R2	-0.3108	0.0531	0.9146	0.9828	0.9798
	NRMSE	1.1449	0.9731	0.2923	0.1311	0.1421
Generalization FL -> SC	MAE	151936.8438	131516.125	19109.916	17394.168	15692.9873
	R2	-0.6068	-0.2168	0.9596	0.9638	0.9707
	NRMSE	1.2676	1.1031	0.201	0.1904	0.1712
Generalization GA -> FL	MAE	437491.4062	44095.7656	44387.4062	53430.0625	68644.3203
	R2	0.2801	0.9919	0.9919	0.9883	0.9839
	NRMSE	0.8485	0.0901	0.0897	0.1083	0.1271
Generalization GA -> SC	MAE	110432.5	14388.5625	13461.3564	13739.5166	14401.3418
	R2	0.1196	0.9806	0.9821	0.9843	0.9798
	NRMSE	0.9383	0.1392	0.1339	0.1254	0.142
Generalization SC -> FL	MAE	424390.75	47684.0977	90427.2734	84410.2734	83852.9453
	R2	0.3208	0.9903	0.9352	0.968	0.9719
	NRMSE	0.8241	0.0984	0.2547	0.1788	0.1676
Generalization SC -> GA	MAE	29812.2109	7731.0142	5788.4355	6306.5957	7316.2769
	R2	0.4087	0.9602	0.9809	0.97	0.9542
	NRMSE	0.769	0.1995	0.1382	0.1732	0.214