DevelopmentODE: Structured Neural ODEs for Early Brain Development Dynamics Across a Decade

Kaiqiao Han; Haitao Chen; Bryan Quah; Xiaoda Wang; Janelle Liu; John H Gilmore; Wei Gao; Yizhou Sun

cs.LGSep 27, 2026

DevelopmentODE: Structured Neural ODEs for Early Brain Development Dynamics Across a Decade

Authors: Kaiqiao Han, Haitao Chen, Bryan Quah, Xiaoda Wang, Janelle Liu, John H Gilmore, Wei Gao, Yizhou Sun

Organizations: University of California, Los Angeles · Cedars-Sinai Medical Center · Emory University · University of North Carolina at Chapel Hill

Abstract

Understanding how individual brain development unfolds over childhood requires modeling developmental trajectories from sparse longitudinal observations. Long-term neurodevelopmental forecasting is challenging because each child is typically observed at only a few irregularly spaced visits, while developmental dynamics vary across individuals and age. Generic continuous-time models accommodate irregular timing but often absorb these factors into a single flexible transition function, providing little structure for how population progression, individual variability, and developmental age shape the dynamics. We propose DevelopmentODE, a structured continuous-time framework that organizes population- and subject-specific variation within a shared developmental geometry while allowing the governing dynamics to evolve with age. The model builds this geometry around a developmental canal representing the population trajectory, whose local direction provides a reference for organizing subject-specific variation. Subject deviation velocities are constrained relative to this direction, while a shared nonlinear deviation field captures individual developmental motion without disrupting population-level progression. DevelopmentODE further models developmental non-stationarity through ordered age-dependent deformations of the shared vector field, progressively adapting a common dynamical structure as age changes, while elapsed time determines the integration horizon. This formulation uses population-level developmental structure to guide learning from sparse individual trajectories while allowing dynamics to evolve smoothly with age. We evaluate DevelopmentODE on longitudinal fMRI by predicting future functional connectivity of the same child from earlier observations. DevelopmentODE consistently outperforms competing baselines across short- and long-horizon predictions.

Figures & tables

Figure 1: Overview of DevelopmentODE. Sparse and irregular longitudinal FC observations are mapped into a shared developmental geometry, where DevelopmentODE models continuous trajectories and predicts future brain states at arbitrary target ages. Latent dynamics are organized around a population developmental canal $\mu_{\eta}(t)$ , with subject-specific evolution modeled as deviation motion in the complementary geometry (bottom left). Smoothly ordered age-dependent modulation adapts the shared deviation field across development (bottom right).

Model	Short-term				Long-term				All
	MSE $\downarrow$	MAE $\downarrow$	$R^{2}$ $\uparrow$	PCC $\uparrow$	MSE $\downarrow$	MAE $\downarrow$	$R^{2}$ $\uparrow$	PCC $\uparrow$	MSE $\downarrow$	MAE $\downarrow$	$R^{2}$ $\uparrow$	PCC $\uparrow$
RNN	3.07	1.23	34.51	59.36	3.10	1.23	31.65	57.19	3.09	1.23	32.80	58.05
GRU	3.09 -0.4%	1.23 0.0%	34.29 -0.6%	59.18 -0.3%	3.10 -0.1%	1.23 +0.1%	31.64 -0.1%	57.18 0.0%	3.10 -0.2%	1.23 +0.1%	32.69 -0.3%	57.97 -0.1%
LSTM	3.07 0.0%	1.22 +0.3%	34.59 +0.2%	59.40 +0.1%	3.09 +0.4%	1.22 +0.6%	31.96 +1.0%	57.43 +0.4%	3.08 +0.3%	1.22 +0.5%	33.01 +0.7%	58.22 +0.3%
Transformer	3.04 +1.3%	1.22 +0.6%	35.32 +2.3%	60.02 +1.1%	3.02 +2.5%	1.21 +1.4%	33.42 +5.6%	58.47 +2.3%	3.03 +2.0%	1.21 +1.1%	34.17 +4.2%	59.09 +1.8%
Direct MLP	3.03 +1.5%	1.22 +0.2%	35.58 +3.1%	60.09 +1.2%	3.07 +1.1%	1.23 -0.1%	32.41 +2.4%	57.66 +0.8%	3.05 +1.2%	1.23 0.0%	33.68 +2.7%	58.63 +1.0%

Table 1: Evaluation results on short-term, long-term, and all prediction pairs. Metrics include MSE (

\downarrow

), MAE (

\downarrow

R^{2}

(

\uparrow

), and PCC (

\uparrow

). The values following each metric denote the relative improvement (%) over the RNN baseline, where positive values indicate better performance according to the direction of each metric. The best results are in bold , and the second-best are underlined . MSE,

R^{2}

, and PCC are multiplied by 100, and MAE by 10, for readability.

Variant	MSE $\downarrow$	PCC $\uparrow$
Ours	2.939	60.51
w/o Canal	2.975 (+1.22%)	59.96 (-0.91%)
w/o Time	2.997 (+1.97%)	59.58 (-1.54%)
w/o Stage	2.997 (+1.97%)	59.61 (-1.49%)

Table 2: Ablation study on all test pairs. Relative changes from the full model are shown in parentheses.

Figure 2: Forecasting error under different amounts of supervision.

Panel of Figure 3 — Figure 3: Temporal analysis of longitudinal FC forecasting. (a) Prediction performance as a function of the source–target temporal interval. (b) Performance across source-time and target-time pairs. (c) Performance across different target time.

Panel of Figure 4 — Figure 4: Robustness of DevelopmentODE to numerical solver, latent representation dimension, and developmental stage-boundary choices.

Model	MSE $\downarrow$	PCC $\uparrow$
DevelopmentODE	4.0467	53.6233
Time-MLP	4.6256	48.6872
Direct MLP	4.6728	48.3735
Age-NODE	5.0071	44.4017
Generic NODE	5.0102	44.3411

Table 3: Comparison on the external BCP cohort. MSE and PCC are multiplied by

100

for readability.

Appendix figures & tables7 assets

Supplementary material from the paper’s appendix.

Appendix

Age	N	$\\|\dot{\mu}\\|_{2}$	$\lambda(t)$	$\mathbb{E}\\|\tilde{r}\\|_{2}$
3 weeks	228	284.007	27.782	11.361
1 year	233	109.156	25.219	13.384
2 years	199	101.897	21.251	13.882
4 years	116	73.443	14.464	12.672
6 years	156	52.219	11.598	12.197
8 years	152	37.022	10.192	13.739

Appendix

Table 4: Developmental geometry across age on held-out subjects. Values are averaged over five subject-disjoint folds.

Transition	N	Population	Deviation
3 weeks $\rightarrow$ 1 year	159	0.530	0.470
3 weeks $\rightarrow$ 2 years	109	0.489	0.511
3 weeks $\rightarrow$ 4 years	71	0.022	0.978
3 weeks $\rightarrow$ 6 years	65	0.033	0.967
3 weeks $\rightarrow$ 8 years	90	0.040	0.960
3 weeks $\rightarrow$ 10 years	76	0.106	0.894

Appendix

Table 5: Population-aligned and complementary components of observed longitudinal changes on held-out subjects.

Transition	N	$\bar{\lambda}$	$\log\frac{\\|\tilde{r}_{b}\\|_{2}}{\\|\tilde{r}_{a}\\|_{2}}$
3 weeks $\rightarrow$ 1 year	159	26.693	0.181
1 year $\rightarrow$ 2 years	130	23.344	0.038
2 years $\rightarrow$ 4 years	54	17.215	-0.073
4 years $\rightarrow$ 6 years	53	12.813	-0.056
6 years $\rightarrow$ 8 years	82	10.715	0.106
8 years $\rightarrow$ 10 years	87	9.942	0.004

Appendix

Table 6: Learned age-dependent modulation and observed deviation persistence for adjacent developmental transitions.

Model	Short-term								Long-term								All
	MSE	$\downarrow$	MAE	$\downarrow$	$R^{2}$	$\uparrow$	PCC	$\uparrow$	MSE	$\downarrow$	MAE	$\downarrow$	$R^{2}$	$\uparrow$	PCC	$\uparrow$	MSE	$\downarrow$	MAE	$\downarrow$	$R^{2}$	$\uparrow$	PCC	$\uparrow$
RNN	3.06	–	1.22	–	34.78	–	59.61	–	3.17	–	1.24	–	29.96	–	56.00	–	3.13	–	1.23	–	31.88	–	57.44	–
GRU	3.06	-0.09	1.22	0.13	34.75	-0.09	59.59	-0.02	3.20	-0.64	1.24	-0.21	29.51	-1.48	55.71	-0.53	3.14	-0.42	1.23	-0.07	31.60	-0.87	57.26	-0.32
LSTM	3.06	0.01	1.22	0.24	34.81	0.10	59.63	0.03	3.18	-0.21	1.24	0.09	29.81	-0.49	55.93	-0.13	3.13	-0.12	1.23	0.15	31.81	-0.22	57.41	-0.06
Transformer	3.04	0.79	1.22	0.24	35.28	1.46	60.01	0.68	3.06	3.69	1.21	2.02	32.63	8.94	57.86	3.31	3.05	2.56	1.22	1.32	33.69	5.68	58.72	2.23
Direct MLP	3.01	1.66	1.22	0.35	35.92	3.28	60.36	1.26	3.13	1.53	1.24	0.22	31.07	3.73	56.70	1.24	3.08	1.58	1.23	0.27	33.01	3.54	58.16	1.25

Appendix

Table 7: Evaluation results when models are trained only on adjacent within-subject transitions and evaluated on all forward test pairs. Metrics include MSE (

\downarrow

), MAE (

\downarrow

R^{2}

(

\uparrow

), and PCC (

\uparrow

). The change rate (%) is computed relative to the RNN baseline. The best results are in bold , and the second-best are underlined .

Model	Short-term								Long-term								All
	MSE	SD	MAE	SD	$R^{2}$	SD	PCC	SD	MSE	SD	MAE	SD	$R^{2}$	SD	PCC	SD	MSE	SD	MAE	SD	$R^{2}$	SD	PCC	SD
RNN	3.07	0.064	1.23	0.011	34.51	0.488	59.36	0.354	3.10	0.040	1.23	0.008	31.65	0.804	57.19	0.585	3.09	0.039	1.23	0.007	32.80	0.562	58.05	0.395
GRU	3.09	0.064	1.23	0.010	34.29	0.510	59.18	0.379	3.10	0.037	1.23	0.004	31.64	0.723	57.18	0.531	3.10	0.036	1.23	0.004	32.69	0.483	57.97	0.342
LSTM	3.07	0.067	1.22	0.010	34.59	0.450	59.40	0.346	3.09	0.035	1.22	0.005	31.96	0.881	57.43	0.628	3.08	0.037	1.22	0.005	33.01	0.547	58.22	0.381
Transformer	3.04	0.070	1.22	0.011	35.32	0.530	60.02	0.387	3.02	0.036	1.21	0.007	33.42	0.854	58.47	0.601	3.03	0.040	1.21	0.007	34.17	0.607	59.09	0.408
Direct MLP	3.03	0.062	1.22	0.008	35.58	0.421	60.09	0.317	3.07	0.035	1.23	0.005	32.41	0.806	57.66	0.583	3.05	0.038	1.23	0.004	33.68	0.545	58.63	0.401

Appendix

Table 8: Evaluation results on short-term, long-term, and all prediction pairs. Metrics include MSE (

\downarrow

), MAE (

\downarrow

R^{2}

(

\uparrow

), and PCC (

\uparrow

). The SD columns report the standard deviation. The best results are in bold , and the second-best are underlined .

Metric	RNN	GRU	LSTM	Transformer	Direct MLP	Time-cond. MLP	Generic NODE	Age-cond. NODE	MoE NODE	Stage NODE	DevelopmentODE
MSE $\downarrow$	4.7677	4.8221	4.8894	4.7032	4.6728	4.6256	5.0102	5.0071	5.0075	5.0134	4.0467
PCC $\uparrow$	46.3742	46.1490	46.0146	47.0882	48.3735	48.6872	44.3411	44.4017	44.3976	44.5096	53.6233

Appendix

Table 9: Comparison on the external BCP cohort. MSE and PCC are multiplied by

100

for readability. The best results are in bold , and the second-best are underlined .

Pairs	Model	$\alpha$	MSE $\downarrow$	$R^{2}$ $\uparrow$	MAE $\downarrow$	PCC $\uparrow$
All	Fixed comp.	$0$	$2.9403\pm 0.0326$	$36.128\pm 0.509$	$1.2009\pm 0.0046$	$60.495\pm 0.383$
All	Learnable aligned	$0.153\pm 0.088$	$2.9390\pm 0.0321$	$36.158\pm 0.527$	$1.2005\pm 0.0052$	$60.515\pm 0.396$
Adjacent	Fixed comp.	$0$	$2.9451\pm 0.0262$	$36.534\pm 0.321$	$1.2033\pm 0.0041$	$60.773\pm 0.256$
Adjacent	Learnable aligned	$0.198\pm 0.025$	$2.9428\pm 0.0266$	$36.586\pm 0.307$	$1.2032\pm 0.0042$	$60.808\pm 0.242$

Appendix

Table 10: Sensitivity to a learnable population-aligned component. Results are mean

\pm

SD over five subject-disjoint folds. MSE,

R^{2}

, and PCC are

\times 100

; MAE is

\times 10

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