Temporally Detailed Hypergraph Neural ODE for Disease Progression Modeling
Authors: Tingsong Xiao
Yao An Lee
Zelin Xu
Yupu Zhang
Zibo Liu
Yu Huang
Jiang Bian
Jingchuan Guo
Zhe Jiang
Venue: ICLR 2026 Poster (2026)
Keywords: Disease Progression Modeling
Neural ODE
Temporally Detailed Hypergraph
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)

Abstract: Disease progression modeling aims to characterize and predict how a patient's disease complications worsen over time based on longitudinal electronic health records (EHRs). For diseases such as type 2 diabetes, accurate progression modeling can enhance patient sub-phenotyping and inform effective and timely interventions. However, the problem is challenging due to the need to learn continuous-time progression dynamics from irregularly sampled clinical events amid patient heterogeneity (e.g., different progression rates and pathways). 
Existing mechanistic and data-driven methods either lack adaptability to learn from real-world data or fail to capture complex continuous-time dynamics on progression trajectories. To address these limitations, we propose Temporally Detailed Hypergraph Neural Ordinary Differential Equation (TD-HNODE), which represents disease progression on clinically recognized trajectories as a temporally detailed hypergraph and learns the continuous-time progression dynamics via a neural ODE framework. TD-HNODE contains a learnable TD-Hypergraph Laplacian that captures the interdependency of disease complication markers within both intra- and inter-progression trajectories. Experiments on two real-world clinical datasets demonstrate that TD-HNODE outperforms multiple baselines in modeling the progression of type 2 diabetes and related cardiovascular diseases.

Published as a conference paper at ICLR 2026
TEMPORALLY DETAILED HYPERGRAPH NEURAL ODES FOR DISEASE PROGRESSION MODELING
Tingsong Xiao
Yao An Lee
Zelin Xu
Yupu Zhang'
Zibo Liu
Yu Huang?. 2.3
Jiang Bian?.3
Jingchuan Guo
Zhe Jiang'
1 DDepartment of Computer and Information Science and Engineering, University of Florida
"Regenstrief Institute
3 *Department of Biostatistics and Health Data Science, Indiana University
*Department of Pharmacy Practice, Purdue University (iiootingoong, zhe.jiang) @ufl.edu
ABSTRACT
Disease progression modeling aims to characterize and predict how a patient's s disease complications worsen over time based on longitudinal electronic health records (EHRs). For diseases such as type diabetes, accurate progression mod- eling can enhance patient sub-phenotyping and inform effective and timely inter- ventions. However, the problem challenging due to the need to learn continuous- time progression dynamics from irregularly sampled clinical events amid patient heterogeneity (e.g., different progression rates and pathways). Existing mecha- nistic and data-driven methods either lack adaptability to learn from real-world data or fail to capture complex continuous-timg dynamics on progression trajec- tories. To address these limitations, we propose Temporally Detailed Hypergraph Neural Ordinary Differential Equation (TD-HNODE)) which represents disease progression on clinically recognized trajectories as temporally detailed hyper- graph and learns the continuous- time progression dynamics via a neural ODE framework. TD-HNODE contains a learnable TD-Hypergraph Laplacian that cap- tures the interdependency of disease complication markers within both intra- and inter-progression trajectories. Experiments on two real-world clinical datasets demonstrate that TD-HNODE outperforms multiple baselines in modeling the progression of type diabetes and related cardiovascular diseases.
INTRODUCTION
Many chronic diseases follow progressive trajectory with multiple complications that worsen over time (Uddin et al. 2023). For example, one patient with type 2 diabetes may gradually develop retinopathy, which advances to visual impairment and eventually to blindness. Another patient may develop hypertension, then atrial fibrillation, and ultimately heart failure. Other patients may experi- complications both trajectories. The goal of disease progression modeling is learn how patient's complications evolve over time from longitudinal electronic health records (EHRs) (Mould) 2012 Wang et al. 2014] Xiao et al. (025a).. Specifically, each clinical visit is associated with a risk factor feature vector (e.g.. lab results, medications) and complication marker vector (e.g., presence of hypertension, atrial fibrillation,
cerebrovascular disease). The task to predict the compli- cation marker vector at the next visit, with the constraint that marker progression follows set of clinically verified trajectories. Disease progression modeling plays a crucial role patient sub- phenotyping (i.e., grouping patients into categories based on heterogeneous progression patterns) and informing effective and timely treatment (Buil-Bruna al. 2015 Prague et al.) 2013).
However, the problem poses several technical challenges. First, patient records of hospital visits are often sampled at irregular time points, although the underlying disease conditions evolve in contin- uous time. Second, many chronic and progressive diseases-such as type diabetes, Alzheimer's disease, chronic kidney disease (CKD). cancer (e.g.. breast or prostate), and cardiovascular dis- eases have clinically recognized progression trajectories that are routinely used to guide treatment planning. Incorporating these validated pathways into a data-driven model is non-trivins con-tiivial, as it re- quires structured representation that encode multi-step, high-order progression dependencies

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Published as a conference paper at ICLR 2026
beyond pairwise relations. Third, the progression dynamies heterogeneous among patients, as reflected by the varying progression rates and progression trajectories (e.g., some patients rapidly de- velop kidney damage while others remain stable for many years, and some patients may experience neuropathy followed by a foot ulcer).
Existing works disease progression modeling can be broadly categorized into mechanistic and data-driven approaches (Cook and Bies. 2016 Mould 2012). Mechanistic models (van Schaick ct al., 2015) |Sahhar 1995) incorporate biological. pathophysiological. and pharmacological pro- cesses into the modeling structure, providing enhanced interpretability but with limited adaptability to real-world data. Data-driven approaches can be further divided into traditional machine learning and deep learning. Traditional machine learning methods often use hidden Markov models (Jack- son et al. 2003 Sukkar et al., 2012 Liu et al. 2015) to capture transition probabilities between disease stages, but they typically rely on strong assumptions about data distribution at known pro- gression stages and cannot learn implicit stages based on complex feature representations. As large- scale EHRs have become increasingly available over the last decade, deep learning methods have been widely developed (Shickel et al.] 2017, Solares et al., 2020), including recurrent neural net- works (e.g.. LSTM) (Zhang] 2019; Zhang al.] 2019; Sohn et al., 2020), attention- based models (e.g., Transformers) (Zhang | Zisser and Aran [024] Xiao et al. [022bb] (Shmatko et al.
2025
Song et al.] (2025), and Neural Ordinary Differential Equations (Neural ODEs) (Chen et al.
2018
Goyal and Benner 2023] Chen et al. 2024a). Neural ODE models can capture the continuous-
time dynamics of disease progression based on urregular-tmme clinical events, but existing models (Qian et al.) 2021; Dang et al., (023) fail to incorporate the clinically verified progression path-
ways. Continuous-time graph neural networks (Rossi et al. 2020 Tian al. 2021 Liu al. 2024 Cheng et al. 2024) can potentially represent known progression trajectories as a directed graph, where nodes are complication markers and temporal edges represent progression between them, but a normal graph only captures pairwise interactions (between one complication and its immedi- ate predecessor or successor) and thus misses high-order interactions across all complication nodes along trajectory (pathway) (Yoon al.) 2020).
To address these limitations, we propose the Temporally Detailed Hypergraph Neural Ordinary Differential Equation (TD-HNODE), which represents disease progression clinically recognized trajectories as a temporally detailed hypergraph and learns the continuous-ting progression dynam- ies via neural ODE framework. The key idea is follows. We adopt Neural ODE to model continuous time disease progression from imregular-time patient records, where the temporal gra- dient of disease states is governed by hypergraph Laplacian that captures high-order interactions among complication markers along clinically known progression trajectories. A standard hyper- graph Laplacian, however, is static and fails to reflect patient-specific temporal dynamics. To this end, TD-HNODE introduces a learnable TD-Hypergraph Laplacian that contains two novel compo- nents: (1) an attention- based incidence matrix that assigns time-aware, patient-specifie importance to each marker within trajectory (i.e., intra-trajectory dynamics), and (2) learnable hyperedge weight matrix that captures dependencies across different trajectories (i.e., inter-trajectory corre- lations). It is worth noting that our proposed framework differs from existing temporal hypergraph neural networks (Lee and Shin 2023 Liu et al. [022]] Lee and Shin 202T), which assign timestamps at the level of entire hyperedges, failing to capture the fine-gre fine-graieed temporal progression details within each hyperedge. Although some methods adopt recurrent modules (e.g.. RNNs (Wang et al. 2024a Younis and Ahmadi (024)) and others incorporate continuous-time dynamics (e.g., Neural ODES (aao et a 2023 Luo et al. 2023)) ), they still construct hypergraph snapshots over discrete intervals, where hyperedges remain static units without modeling marker-level timestamps. We vali- date TD-HNODE on two real-world EHR datasets, showing consistent improvements over baselines in modeling the progression of type 2 diabetes and cardiovascular diseases.
PROBLEM STATEMENT
2.1 PRELIMINARIES
A patient's medical history can be represented as a sequence of hospital visits called encounters. We denote encounter for patient at time tx (x),(t))) where k denotes the index of the k-th encounter (timestamp) in the patient's encounter sequence, and x,, (tx) e R is a vector of risk factors (c is the number of risk factors), such medications, laboratory test results, and
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(1) Markers
(b)DAG
(d) Problem Example
4
4
4
4
4
HP
AF
HF
P1
HP
HP
1
CD
S
P2
AF
AF
(o)yD-Hyyergraph
Medications
Ingut
ID-HNODE
Output
HF
roulls
CD
HF
ei
-
Vital signe
Input
0 1
HP
Risk Factors
(t)),... X. (tg)
Predicted y (fa)
(b) Y. (8))
7
CD
62
Figure I: (a) Patient u's marker status vector from t to is; (b) The corresponding DAG of trajecto- ries Pı and P2; (c) Patient u's HDpperrgga with two temporally detailed hyperedges ex e²; (d)
An example of problem definition showing TD-HNODE's input and output.
vital signs, and Yu(44) (0, 1) is vector of disease complication markers for presence, for absence), such as hypertension (HP), atrial fibrillation (AF). heart failure (HF), cerebrovascular disease (CD), and stroke (S). The set of all n markers is denoted as {v; =......,77 n).
The marker vector (t) evolves over time in the encounter sequence of patient u, reflecting the progression of disease states. How the markers evolve follows set of disease progression trajec- tories (or pathways) that can be constructed based on clinical knowledge. Formally, a trajectory is defined as an ordered sequence of distinct markers, denoted as Pj =< 1 12. ..., ''ps >, where
o! V represents the i-th marker in the j-th trajectory, and |p, is the number of markers.
Figure aa) provides an example of the temporal records of a patient's five disease complication markers (HP, AF, HE, CD, and S) from l to fs. Figure(Tīi) shows two clinically known progression traeectories (in green and yellow) that these markers follow, ie.. P1 =< HP,AF, HF > and P2 =< HP, CD,S > The fact that the patient's 's markers follow these trajectories is highlighted by the red circles as well as the green and yellow arrows in Figure [a]. For instance, once the patient has hypertension (HP= 1) at time (1, the status will persist. The patient's disease progresses from hypertension to cerebrovascular disease (C'D = 1) at time iz and to stroke (S = 1) at t,, following the yellow trajectory. Similarly, the discase progresses from hypertension to atrial fibrillation ta (AF = 1) following the green trajectory. Note that the patient's 's disease complication markers have not yet progressed to heart failure (HF) by 1s, but could happen a future time.
We assume that the disease markers in the trajectory are irreversible. i.e., the binary status of mark- ers can only transit from 0 to (or stay the same) within an encounter sequence. This assumption is reasonable for many chronic diseases-such as type 2 diabetes, Alzheimer's disease, and chronic kidney disease-ther ttht exhibit irreversible progression (e.g., organ fibrosis, vascular damage) (Wu
al. 2021 Kazemian et al. 2019 Bhatwadekar et al. 2021 Wang al. 2024b)
The set of clinically known progression trajectories forms a Directed Acyclic Graph (DAG), as illustrated in Fi). To capture the higher-order marker patterns within trajectories, we propose to represent trajectories using a hypergraph (Gallo et al., (993). Specifically, we represent the set of hyperedge. There are two key advantages: (1) hyperedge
markers within the same trajectory can connect multiple markers in an entire trajectory (pathway), enabling the modeling of high- order dependencies beyond pairwise relations (Feng al., 2019 (0119/Gn Gao al. 2022): ; and (2) different
hyperedges overlap with each other through common markers (potentially pivotal nodes) (Chitra and Raphael 2019), making easier to model interdependency across progression pathways.
We formally define the disease progression hypergraph as H = (V,E), where V is the set of markers, also referred to nodes throughout the paper, and & denotes the set of hyperedges, each corresponding to predefined trajectory (we use the terms 'hyperedge', 'pathway', and 'trajectory' interchangeably). Specifically, E = {c1, (ei,e,,.... . Cm and 6j (v), ",...,, "/c, 1. where m is the total number of predefined trajectories, and leyl the number of markers the j-th trajectory.
For lab-derived markers such as "HbAIc High/Low" and "Poor Lipid/BP" we use their first occurrence as a progression signal in y. indicating that the patient's chronic stage has reached a clinically significant level (e.g., once patient has experienced "Poor Lipid," it signals progression to that stage regardless of later fluctuations). This design choice was confirmed by clinician (M.D.) among co-authors. Meanwhile, the fluctuating lab values (e.g., GFR, HDI., Triglycerides) are incorporated through the risk factor features x.
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To capture each patient's actual progression, we define Temporally Detailed Hypergraph (TD- Hypergraph) HD = (V, E"), where Ex {er,ey ci...., c is a set temporally detailed hy- peredges. Each hyperedge records both the markers and the timestamps which they were first observed for patient u: ej {(0] (1), (r2
(v g fk),; f2)....,
(1241,00),,-- .o.............................. (v/ |e,1> o0)), where (o)),
markers up oi have been observed (with t, being the timestamp of first occurrence), and the placeholder oo indicates markers not yet observed. Since patient may not develop all markers in a predefined trajectory, the observed portion can be shorter than the full trajectory. For example, as shown in Figure Ta). the patient's s observed progression up to i5 yields two temporally detailed trajectories: pf
(HP, 41, (AF, F, (a) > (shorter than the full trajectory pl =< AF, HF >) and p2 =< (HP, 1),, (CD, 12); (S, (4) >. Figure (c) shows the resulting TD-Hypergraph at time ts: the markers have progressed to (stroke) but not HF (heart failure) yet. The TD-Hygerggaph lime-dependent-it evolves complication markers progress time, reflected by timestamp (temporal details) updates on hyperedges.
A list of commonly used notations is provided in Table (Appendix A
2.2 PROBLEM DEFINITION
Given patient encounter data with risk factors (t)( ..... .,. x (tx))==1 target complication mark- ers (ttt....... ........yyttt yu
and the TD- Hyyyggrgga {7²)) } the problem aims to learn TD-HNODE model Se such that:
fu(t++1) fe (x.(tt)....xx((tt)) yu(ti), ... (t();; H²).
The objective is to minimize the binary cross-entropy loss:
min N (f yu(ta+1)).
Figure [d) provides an illustrative example. The TD-HNODE model takes as input patient u's risk
factors (t,, -.... Xr, (fg) (e.g., medications, lab results, and vital signs, Figure 1 (() left), marker status (f. Y. (ts) (Figure [a)), and a TD-Hypergraph H. (Figure[c)) The model predicts future marker status at ig (Figure d) right).
METHODOLOOGY
An overview of our model framework: As shown in Figure 2a., TD-HNODE employs Neural ODE model continuous-tims disease progression from irregular-time encounters, where nodes represent complication markers and hyperedges represent clinically known progression trajectories (patient index u omitted for simplicity). At each encounter time lg, risk factors x(4x) and compli- cation markers y(&)) are embedded into node representations of the TD- TD-Hypergrapp H". which used construct learnable TD-Hypergraph Laplacian L(t) (Figure b)) capturing intra-trajectory dynamics and inter- trajetrrajectory dependencies. Let S(t) E R denote the hidden state of markers at time f. where n is the number of markers and d is the embedding dimension. The Laplacian L(t), together with risk factors and hidden state Stt,), is passed the Neural ODE solver to update the latent state S(t+ which is then decoded to predict marker status ý (tx+1).
3.1 NEURAL ODE AND HYPERGRAPH NEURAL NETWORK
General Neural ODE. Neural ODE learns continuous dynamic function S(t):
d(t(t)
dt / - (1,,()),x(())))
where are learnable parameters and f(-) models the temporal gradient of the disease state, The
hidden state is initialized as S(t,) 0 Rxd with the patient's 's initial conditions introduced through the risk factor embedding (x(())) Eq.
Static hypergraph Laplacian. We instantiated f(.) using hypergraph Laplacian L (Feng et al. 2019) to enable multi-way message passing over disease trajectories:
((1,
[S(() h (x(())) e,
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(a) Overview TD-HNODE
TD-Hypergraph Laplacian
S(4x)
L=1- D. W,D, H,
Hidden State
S(6b 1)
8(1))
TD-Hypeggrahh
Laplacian
II Attention-base based Incidence Matrix
W Laamable Hyperedge Weight
Neural ODE Solver
Neural Solver
Decoder
((t-))
I(t,)
Predict
TD-Hypergraph Laplacion
TD-Hyporgraph Laplacian
(1
Cross Attention
PE
PB:
O0
H
fooooo
H
BCE Loss
(t
((,)
(
x(,
(c) Cross Attention
t
t
ej
V)
Figure 2: (a) Overview of the TD-HNODE, with example of patient u's encounter sequence; (b) The TD-Hypergraph Laplacian module combining an Attention- Incidence Matrix and a Learnable Hyperedge Weight Matrix; (c) An illustration of cross attention within hyperedge ej.
where h(-) maps x(t) into the same space as Stt), and O E IRd is learnable transformation matrix. The negative sign simulates diffusion-lie like propagation among markers (Ji et al. 2022). The common hypergraph Laplacian defined as:
- -IID-VV'HDD HWD, 'H "DD,
(5)
Intuitively, this Laplacian governs how information diffuses among markers: markers connected by the same hyperedge (trajectory) exchange signals, with normalization ensuring balanced influence regardless of node or hyperedge size. Here, E RR is an identity matrix, H (0, 1 the incidence matrix with Hi.
if marker v; belongs to hyperedge ej. and H j) 0 otherwise. W Rmxm a diagonal hyperedge weight matrix. The node and hyperedge degree matrices are (i, = E, H(i,j) W(.J),, and De(j.) E. H(i,j).
Limitation. However, the above static Laplacian treats all markers equally within each hyperedge (H is binary) and uses fixed diagonal weights (W), failing to capture: (1) time-varying marker importance within trajectory, and (2) inter-trajectory dependencies due to shared markers and correlated temporal dynamics. We next extend L to learnable, temporally adaptive L() (Eq. 13
3.2 LEARNABLE TD-HYPERGRAPH LAPLACIAN
We proposed two key enhancements: (1) Attention-based Incidence Matrix: we replaced the binary incidence matrix with learnable attention mechanism, allowing the model to assign time-aware, patient-specific importance to each marker node within progression trajectory; (2) Learnable Hy- peredge Weights: instead of assigning fixed weights to hyperedges, we introduced learnable weights that captured trajetrrayc dependencies based on shared markers and correlation patterns. This design allowed TD-HNODÉ perform high-order message passing guided by clinical knowledge while adapting to serieent-speiffie disease trajectories.
3.2. IITENTION-BASED INCIDENCE MATRIX
To account for temporal dynamics and the varying importance of markers in trajectory, we de- signed an adaptive incidence matrix based on cross-attention within each temporally detailed hyper- edge. For a given time we define tx, as the timestamp of the patient's 's most recent encounter before t, and the model uses all sequential encounter data up to tho (i.e., encounters ty for k S ko) to construct the observed progression trajectory and the Ty-Hypergraph Laplacian. Given a tem- porally detailed hyperedge e, = {(]] f1), (Nz, .2).... (v2 (,.),, (v% oo),... (, (efe,) oo)). we denoted the current progression point at time Lx as Ukg omitting the trajectory index for notational
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simplicity. Intuitively, the current progression point divides each trajectory into markers the patient has already developed and markers that may develop in the future. Based on this, we split ej into two subsets: a past set O = {V1,... ..., Uko } containing already observed markers, and a potential set JF (1 containing markers not yet observed.
Each marker v; E Ej had initial embedding b; RR obtained by applying learnable multilayer perceptron to the one-hot encoding of marker vç Since observed markers have real timestamps while future markers do not, we used different positional encodings for each group: continuous-
encoding for observed markers and discrete inde--based encoding for future markers. Specifically, ((i) Piiee (t:) if U; 0², and (ii) Oux (t) if FF, where @rime (t,) I was continuous- time encoding (Xu et al. 2020), and Oiax (i) E IR" was discrete index-based encoding /Vaswani jtt al. 2017).
We computed the query vectors as:
q: time (b; Pume (umeett)) (t:)) if ef
(6)
q: idx (b, + Pixx W ef
where Wo E Rdxd is a learnable projection matrix. In other words, each query encodes "where the patient currently is" in trajectory, combining the marker's identity with its temporal posi- tional context. Key and value vectors were constructed analogously, each using its own learnable projection matrix (WK K. Wy Rdxd) and the same time/index positional encodings.
We then computed the attention weight from the current progression point Uko to each other marker e/ E ej. separately for observed and unobserved markers using their respective encodings. We first define the softmax normalization terms over the past and potential sets:
Zo =
Σ Ieos
time exp Ako
k) time /vd), ZF= = Σ Liess exp(ck
kid /Vd).
(8)
The attention weight is then:
cxpp(
Ako
time
kime
/va)
Zo
if E
a1 (o) =
(9)
exp
ids
k.th /Va)
Zz if
Intuitively, (i Ao) measures how much the current progression point Uky should attend to marker ey using awee- encodings for already-observed markers and position- position-based encodings for markers that may develop in the future. We then constructed the adaptive incidence matrix Hp by modulating each entry with the cross-attention weight from the current marker uk, to the marker ny
H. (i.j) H(i,)) ai(i,k)) if E Ej,
(10)
otherwise.
Here, (i, ko) encodes the directional and time-aware importance of marker v under the current progression context. This formulation allows the incidence matrix to capture both structural relations and temporal dynamics, reflecting the evolving role of each marker during disease progression.
3.2.2 LEARNABLE HYPEREDGE WEIGHT MATRIX
Traditional hypergraph-baat based methods typically assume fixed hyperedge weight matrix W Rmxm, where m is the number of hyperedges. However, this assumption fails to capture vari- able correlation strengths in patient-specific progression across trajectories. To address this, we introduced learnable hyperedge weight matrix W, m based hyperedge representation, which modeled dynamic dependencies among trajectories. The key idea was to derive trajectory- level embeddings from their constituent markers and compute trajectory similarity in learned latent space.
For each marker e; eý, computed its context-enhanced representation v, using self-attention within the subset it belongs to. Specifically, self-attention was performed separately over the past set O' and the potential set FJ, producing v; SelfAutn(v.. O²) if U; E O2, and V; SelfAttn (v,, Fi)

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if E Fi. Here, SelfAttm (U;:)) denotes standard scaled dot-produet attention with is query and all other markers in the same subset as keys and values. The resulting vector vi E R captured context-specific information and was used to form hyperedge-level representations. Specifically, for hyperedge ej E E, we aggregated the value vectors V, to obtain a trajectory-level representation:
gj = Aggregatt V: 0 E eyh:
(11)
where Aggregate()) a differentiable pooling function, such as average pooling.
We aggregated the trajectory-level embeddings from all hyperedges & into trajectory embedding matrix G g.:... Rimxd, where the j-th row of G, i.e. 8j. encoded the representation of the j-th trajectory (hyperedge). We then projected this matrix into a latent space using trainable
linear transformation: G GWe, where We E Rdxd The latent trajectory embedding matrix G was then used to compute the trajectory correlation matrix, also called learnable hyperedge weight matrix:
Wp - GG E mm
(12)
This learnable matrix Wp captures data-driven similarities between all trajectories, allowing the model to emphasize more relevant progression pathways well their interdependency. For ample, the retinopathy trajectory and the nephropathy trajectory may receive a high correlation weight in W because they share common early markers and often co-progress in diabetic patients.
We combined the adaptive incidence H, and the learnable hyperedge weight matrix W, p to form our Knowledge-Infused TD-Hypergraph Laplacian:
L == I D H. W H.
(13)
where I is the n x n identity matrix. In this way, L. encodes intra-trajectory time-sensitive marker dependencies (through Hp) and inter-trajectory correlations (through Wp). Note that L, Hp, and Wp all depend time through tkg. the most recent encounter before the current time t (as defined in Section 3.2.1 To reflect continuous-time progression, we wrote L as L():: for each ODE in- tegration interval [4x, (R.),, the Laplacian is constructed from all encounters up to la and remains fixed throughout the integration steps within that interval, since no new observations arrive between encounters. Substituting L(t) into Eq. (4) yields our final knowledge-infused disease progression model:
d (t - -(() [S()) + ((x())]
(14)
dl
The complete training procedure and pseudocode are provided in Appendix B while the computa- tional complexity analysis is detailed in Appendix C
EXPERIMENTS
4.1 EXPERIMENTAL SETUP
Datasets. We conducted experiments on two EHR datasets: (1) clinical dataset collected from re- gional medical network affiliated with our institution, referred the University Hospital dataset; and (2) MIMIC-IV (Johnson et al. 2023). a publicly accessible EHR dataset. We extracted 34 risk factors associated with the progression of diabetes and its complications, as detailed in Table 4 (Appendix D.I.. and identified 21 outcome markers of diabetes complications V. as summarized in Table (Appendix [D.2). We also constructed disease progression hypergraph H based on expert- validated clinical pathways provided by our clinical collaborators, as detailed in Appendix D.3 The University Hospital and MIMIC-IV datasets contained 2,415 patients and 902 patient sequences, respectively. More details provided in Appendix D.4]
Baselines. To evaluate TD-HNODE, we compared it with representative baselines across four categories. For Sequential Models, we included T-LSTM (Baytas et al. (207)), which handles irregular visits using time-aware LSTM, and ContiFormer (Chen et al. 2024b), which redefines self-attention over evolving latent trajectories. For Temporal Graph Neural Networks, used

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Table 1: Results (%, average std) of all methods on University Hospital dataset and MIMIC-IV dataset, with the best results in bold.
University Hospital
MIMIC-IV
Methods
Accuracy
Precision
Accurncy
Precision
Fl-score
TESIM
69.2
10.6
25.7
12.8
84.1
17.0
58.2
20.5
ConriFormer
77.2.2..2
12.3
65.4
16.7=0.2
86.2+..1
26.2+0.4
82.
36.5
MegaCRN
70.7++0.1
8.1
64.5+0.8
12.9-0.1
83.4+..6
224+0.3
70.1
30.8=0.4
TGNE
72.410.2
8.6g0.1
75.4.0.3
14.2.0.1
85.0g
23.40.
72.8g0.3
32.120.2
DHSL
72.8+0..1
8.3+0.1
60.5
13.5-0.3
82.9+0.2
22.0+a.2
69.9 69.94o.a
29.8+0.2
HyperTime
74.940.2
9.3
59.020.1
13.9
84.61..2
25.1
71.42002
31.72022
NODE CODE-RNN
72.3.0.1
10.7.0.3
$6.920.1
14.4
84.410.2
19.5...
62.320.
27.2
73.0.0.1
10.0.0.1
61.3
15.0.0.1
85.7.0.1
23.5.0.2
74.520.4
32.5.0.2
TD-HNODE
79.4
14.3
7
20.410.4
87.
31820.
85.7.0.4
42.911.a
Table 2: Ablation study results (%) of TD-HNODE on University Hospital and MIMIC-IV dataset.
University Hospital
MIMIC-IV
Hp
W
Accuracy
Precision
Recall
Fl Fl-score
Accuracy
Precision
Recall
Fl -score
/
V
79.4
14.3
79.3
20.4
87.9
31.8
85.7
42,9
X
/
75.3
12.9
77.0
18.9
86.1
27.7
78.5
36.6
/
76.5
79.2
18.7
86.8
38.5
x
x
73.1
10.6
76.1
15.5
83.0
23.3
73.1
30.8
discrete-time MegaCRN (Jiang al. 2023) and continuous-time TGNE (Cheng et al. 2024)) which model temporal graphs via snapshots and event-based messages, respectively. For Temporal Hy- pergraph Neural Networks, compared with DHSL (Wang al.) 2024a), which fuses hyper- graph convolution with GRU, and HyperTime (Younis and Ahmadi 2024). which builds dynamic hypergraphs with eeTMnennanced hyperedge convolutions. For Neural ODE-based Models, included NODE (Chen et al.) (2018), capturing continuous dynamics in latent space, and CODE- [ 2025), which integrates ODEs with RNNs for time-series modeling. Full base- lines implementation details are provided in Appendix D.6
RNN (Coelho al.
We evaluated performance using Accuracy, Precision, Recall, and Fl-score, emphasizing Recall due to class imbalance and the need to detect early progression. High Recall reflects better identification of true positives, critical for clinical deployment. Source code has been included in the Supplemen- tary Material for reproducibility, and implementation details are provided in Appendix D.5
4.2 COMPARISON ON CLASSIFICATION PERFORMANCE
We compared TD-HNODE with all baselines on the University Hospital and MIMIC-IV datasets, with results summarized in Table TD-HNODE consistently achieved the best performance across all metrics. On University Hospital, outperformed the strongest baseline (ContiFormer) by 2.2% in accuracy and 3.7% in F1-score, and by 1.7% and 6.4% on MIMIC-IV. Compared to non-structural models like T-LSTM, NODE, and CODE-RNN, TD-HNODE showed substantial Recall and gains (e.g. +23.4% Recall over NODE on MIMIC-IV), demonstrating its strength in capturing com- plex disease dynamics through its trajectory-aware hypergraph structure. TD-HNODE also sur- passed temporal structure models such as MegaCRN and TGNE. Notably, it achieved 3.9% and 12.9% Recall improvements over TGNE on University Hospital and MIMIC-IV, respectively. These results confirmed that modeling high-order multi-node interactions via hyperedges offered stronger representational power than traditional pairwise edges, enabling more accurate and clinically mean- ingful early disease prediction with fewer false negatives.
Additional experiments on cardiovascular disease are provided in Appendix D.7 further demon- strating the generalizability of TD-HNODE beyond diabetes.
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0.86
0.86
40
0.84
0.84
0.82
20
0.82
0.80
0.80
0.78
-2
0.78
0.76
0.76
University Hospital
0.74
University Hospital
40
0.74
MIMIC-I
0.72
MIMIC-IV
60
64
128
256
512
4
6
10 12
-00-0020 O 20 40 60 (c) t-SNE Visualization
(a) Embedding Dimension
(b) Number of ODE Steps
Figure 3: (a) Recall of END-E on both datasets with varying embedding dimensions. (b) Recall with varying numbers of ODE steps. (c) I-SNE visualization of 1,690 patients from the University Hospital dataset; CI, C2, and C3 denote Clusters 1, 2, and 3.
Complication Markers Subset Onset Time (Mean in Months)
C1 53.3
49.7 49.2 48.7 46.5 48.1 47.8 47.8 47.4 47.3 47.1 46.8 46.5 44.8 42.9 41.6 36.5
60
55
C2
44.1
46.2
44.0
41.6
44.4 42.9
46.0 45.9 45.4
49.8
43.5
44.6
50.4
41.8 38.0
39.6 41.5
50
C3
54.0
64.3
54.4 52.6
57.0 59.3
54.1 59.1 56.6
62.9
59.9
54.1 58.5
52.3 52.6 50.8 48.6
40
Cardiac
Vision
Depress
Cancer
CHF
Ulcer
Neph
Stroke
Hypo
Vi
CVD
Neuro
AFlb
Obesity
Reti
HTN
Keto
Figure 4: Mean onset time (in months from the first encounter) of each marker across patients in each cluster. Earlier values indicate earlier manifestation faster progression.
4.3 ABLATION STUDY
We assessed the impact of two core components in TD-HNODE: the adaptive incidence matrix Hp and the learnable hyperedge weights W,. p. As shown in Table using H. improved Fi-score from 15.5% to 18.7% without W,- and 20.4% with highlighting the benefit of intra-trajectory attention for time-aware marker importance. Meanwhile, enabling W, boosted recall from 76. 16.1% % to 77.0% and Fl-score from 15.5% to 18.9% even with static H, and yielded 5.4%/5,8% gains on recall/F1 on MIMIC-IV, validating the value of modeling inter-trajectory dependencies.
4.4 SENSITIVITY ANALYSIS
We analyzed TD-HNODE's sensitivity to two hyperparametens: embedding dimension d and the number of ODE solver steps in RK4. As shown Figure (1),, increasing d from 64 128 sig- nificantly improved recall (e.g.. from 0.747 to 0.857 on MIMIC-IV,, but further increases showed diminishing returns or overfitting, so we chose d 128. Similarly. Figure 33)) shows that vary- ing the number of steps from 4 to 12 revealed underfitting at lower values (e.g. 4 or 6), while performance stabilized around 10 steps with minimal gains beyond. These results confirmed TD- HNODE's robustness and the suitability of its default hyperparameter settings for real-world EHR applications.
4.5 CASE STUDY
We conducted a case study to apply the TD-HNODE results for patient progression sub-suenoooe phenotyping. Specifically, we extracted the patient embeddings (prior to the decoder layers) learned from TD- HNODE on the University Hospital dataset. We visualized the patient embeddings in 2D using SNE projection and identified three clear clusters, shown in Fjgure We then used hierarchical clustering to group patients into these three clusters and analyzed disease progression patterns within each cluster. Specifically, for each of the 21 complication markers, (in months since the first encounter) for patients within each cluster. A smaller onset time indicated faster progression of complication outcomes. As shown in Figure 4 patients in Cluster 3 exhibited the slowest progression, followed by Cluster 1, and then Cluster 2, which showed the most rapid progression. For example, compared to Cluster 3, patients in Cluster experienced earlier onset by
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9 months in Cardiac Revascularization (Cardiac), 18 months in Blindness and Vision Loss ( (Vision), and 12 months in Congestive Heart Failure (CHF). These results demonstrated that TD-HNODE effectively captures the heterogeneity within the patient cohort.
CONCLUSION AND FUTURE WORK
In this work, we proposed TD-HNODE, novel framework for continuous- disease progression modeling that integrates medical knowledge with a TD-hypergraph--are -aased Neural ODE. The method captures both intra- and inter-traecctory dependencies, and experiments on real-world EHR datasets demonstrated its effectiveness in modeling diabetes progression. For future work, our framework currently focuses on known disease progression pathways, as in chronic diseases such as diabetes, and should be extended to infer unknown partially characterized trajectories (e.g., via frequent pattern mining or Bayesian networks). In addition, we plan to incorporate causal inference to eval- uate the impact of complex treatment regimens.
ACKNOWLEDGMENTS
This material is based upon work supported by the National Science Foundation (NSF) under Grant No. IIS-2147908, IIS IIS-2207072, OAC-2152085, OAC-2402946, and OAC-2410884.
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APPENDIX
A MATHEMATICAL NOTATIONS
The main mathematical notations of this paper are shown in Table 3
Table 3: Main mathematical notations.
Notation
Description
X₂ (tt)
Risk factors vector of patient u time th Marker status vector of patient u at time lu Set of markers (nodes)
Yu (1a)
The j-th trajectory: < of. ug...... o [p:1
Pj
n
Number of predefined markers (nodes)
m
Number of predefined trajectories (hyperedges)
The i-th marker in trajectory Pj Disease progression hypergraph: (V,
H
Dj
Patient u's temporally detailed trajectory along Pj
Hu
Hyyeggrgga of patient (V,E")
TD- TD-Hyperedge of along Pj
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B TRAINING ALGORITHM
After integrating the ODE from the latest observed timestamp ie to fx+1, we obtain ((4++1) E Rnxd, the hidden representations of all markers at time f These embeddings are then mapped to prediction scores for each marker, followed by a sigmoid activation to estimate the probability of presence for each marker at f The model is trained by minimizing the binary cross-entropy (BCE) loss between the predicted probabilities and the ground-truth marker statuses.
The overall training process is detailed in Algorithm I below. In particular, we adopt an auto- regressive training loop: for each patient, the model integrates the continuous dynamics forward in time, using the hidden state at l to predict marker outcomes at 4 At each step, the TD- Hypergraph Laplacian computed at time iy guides the flow of information, capturing fine-grained trajectory dynamics and marker dependencies.
Algorithm Training procedure of TD-HNODE
Require: Encounter sequences for patients:
(x,( ..... xa(h,,) va(h),... ................................
y. yu (1)m
TD-Hyeergapphs: {Hm)
Ensure: Model parameters Q u=1
1: Initialize all parameters
2: for epoch in MaxEpoch do
3: for each patient u do
4: Initialize hidden state Su(t1)
5: for timestamp t f- f1 : tt., do
6:
Compute incidence matrix (t.) (Eq. 10
7:
Compute hyperedge weights W (1+)) (Eq. 12
8:
Compute TD- HD-Hyyergraph Laplacian L(t,) (Eq.
13
9:
Update state: (,44++1) ODESolver(S. (4+), Xx (lx); L((x), (+14))
10:
Vu(t++1) = (Sense(S
Compute loss C BCE(ý, yu
12: end for
end for
14:
Update O via gradient descent on accumulated loss
15:
end for
16: return +
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C COMPUTATIONAL COMPLEXITY
Assume the average number timestamps per patient sequence I,, the batch size is B, the number of markers (nodes) is n, the number of hyperedges is m, the number of attention heads is h, and the hidden dimension is d. We denote the per-head dimension dA d/h, the average number of markers per hyperedge as ne. and the number of Runge- Kutta steps in Neural ODE 8.
Attention-based incidence matrix (Eq, p With multi-head attention, each head operates dimension dj. The per-head query dimensions are Q E IR Bxxxda and the key dimensions are K E R 1B.xxxn. xdd, The dot-product attention computation across all heads has complexity O(B m Th d) per timestamp.
Hyperedge weight matrix (Eq. 12 Self-attention pooling within each hyperedge costs ( m- .d). The linear projection G G W costs O(B m dd),, and computing the trajectory correlation matrix W, G E Rmxm costs O(B m2 d). Total complexity is O (m n d + m. d² m d)).
TD-Hypergraph Laplacian (Eq. 13 The Laplacian E R is constructed once per timestamp and reused across ODE steps. The dominant costs are the matrix products Hp Wp at O(n m²) and the subsequent product with H at On m). Total construction cost is O(B (n - m² + n² m)) per timestamp.
ODE solver: Each Runge-Kurte Kutta step applies the precomputed L R (S(t) + h(x(t))) E Rnxd, costing O(n² d), and multiplies by the projection matrix @ E Rdxd costing On d²). Over steps, the total cost is O(B -n --n-d- (n + d)) per timestamp.
The overall per-batch complexity of TD-HNODE is
Tioial +(BB L (mmad + mn@d + mdd + m2d d + am? + + n&m + snd
and since ne S n, we can further bound it by
Tomal O(B L. (mn?d d mdd + sn
Given our settings with [20, 30), m E [10. d = 128, and i S 20, the dominant cost is the ODE solver O(B- L. n dd), and overall complexity scales linearly with the sequence length Ż, the number of markers n, and the number of ODE steps s.
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D EXPERIMENTS
D.1 RISK FACTORS
The risk factors were defined below by our clinical co-author (MD) based on established expertise and supported by the literature on diabetes complications (Tomic et al. 2022).
Table 4: List of 34 risk factors used in the study.
SEX_CD
GFR
HDL
Triglycerides
Beta blockers
CCB
DPP4i
Lipid
Loop
Metformin
Non_loop
RAS
Sulfonylurea
Thiazolidinedione
Alcohol use disorder
Angina flag
Cardiovascular Disease
Chronic_kidney kkddney_dssease ,disease
Drug use disorder
End Staget Renal Disease
Exercise
Gestational diabetes
History_of_Mie of__Moocariia_Imag Inffacttion
History _strok
HIV_AIDS
Hypercholesterolaemia
Lower extremity amputation
Myocardial Infarction MI
Organ transplant
Peripheral Peripheral.vasculaa vascular disease
Photocoagulation
Pregnancy
Retinopathy intravitreal injections
Secondary diabetes
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D.2 COMPLICATION MARKERS
We listed a detailed table reporting the positive prevalence (%) for each of the 21 complication markers both datasets (shown below Table The 21 diabetes complication outcome markers were defined by our clinical co-author (MD) based on established clinical expertise and supported by the literature on diabetes complications [Tomic al., 2022). In our formulation, we treat all 21 markers as irreversible. Although some markers such as 'HbAlc Low', 'HbAlc le High', 'Poor Lipid", and 'Poor BP" are laboratory test results that may fluctuate over time, we emphasize their first occurrence as an indication that the patient has progressed to this stage of disease or has experienced this level abnormality. Following clinical guidance, we therefore model the initial onset of each marker significant progression event that remains active in the disease trajectory representation.
Table 5: Complication marker prevalence in University Hospital and MIMIC-IV.
Complication Markers
University Hospital
MIMIC-IV
HbAlc Low
0.058
0.028
Poor Lipid
0.026
0.039
Hypertension (HTN)
0.068
0.043
Obesity
0,058
0,022
Foot Ulcer (Ulcer)
0,005
0,001
Blindness and Vision Loss (Vision)
0.003
0.001
Visual Impairment (VI)
0,001
0.001
Heart Failure (CHF)
0.018
0,024
Nephropathy (Neph)
0.030
0.026
Neuropathy (Neuro)
0.051
0.018
Retinopathy (Reti)
0.011
0,006
Cerebrovascular Disease (CVD)
0.011
0,002
Stroke
0.011
0.003
Depression (Depress)
0.026
0.038
Hypoglycemia (Hypo)
0.022
0,003
HbAle Ic High
0.053
0.018
Poor BP
0.024
0.005
Cardiac Revascularization (Cardiac)
0,009
0,010
Atrial Fibrillation (AFib)
0,007
0,021
Cancer
0.014
0.001
DKA (Keto)
0.002
0.019
Average
0.024
0.016
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D.3 DISEASE PROGRESSION PATHWAYS
The predefined hyperedges (progression pathways) were determined below with guidance from our clinical co-author (MD) and supported by established medical literature (Fonseca
2009
Yu al.
2024).
HbAlc High + Poor Lipid +
Hypertension Poor BP + Atrial Fibrillation + Heart
Failure
HbAlc High -> Obesity
HbAle High + Retinopathy Visual Impairment Blindness and Vision Loss
HbAlc Low + Hypoglycemia
HbAlc High + DKA
HbAle High - Poor Lipid + Hypertension Poor BP - Cardiac Revascularization
HbAle High > Depression
HbAlc High Poor Lipid Hypertension Poor BP
Cerebrovascular Disease
Stroke
HbAle High + Neuropathy + Foot Uleer
HbAle High >> Nephropathy
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D.4 DATASET DETAILS
We provide additional details about the two datasets used in our study.
University Hospital Dataset. This study was approved by the Institutional Review Board (IRB) of our institution. This dataset contains longitudinal EHR sequences for 2,415 diabetic patients collected from regional medical network affiliated with our institution. Each patient record consists of time-ordered sequence of hospital encounters, including structured clinical information such as laboratory test results, vital signs, medications, and diagnosis codes.
MIMIC-IV Dataset. MIMIC-IV (Johnson et al. 2023) is a publicly available EHR dataset col- lected from the Beth Israel Deaconess Medical Center. Although not specifically designed for dia- betes, we identified 902 patients with at least one diabetes-related complication by mapping ICD- 9/10 diagnosis codes our predefined marker V.
Preprocessing. For both datasets, we organized each patient's data as sequence of 20 encounters, with each encounter containing risk factor vector x. (4x) and binary complication marker vector Yu (fx). For the University Hospital dataset, the raw encounter sequence lengths ranged from to 40. To standardize the input format, fixed the sequence length at 20. The basic statistics of both datasets are shown in T For patients with fewer than 20 encounters, we applied padding at the end of the sequence; for those with more than 20, selected the latest 20 encounters, disease progression events tend to occur in later stages of follow-up.
Table 6: Dataset statistics for University Hospital and MIMIC-IV.
Number of Encounters
Time Span (Months)
Dataset
Min
Avg
Max
Min
Avg
Max
University Hospital
10
15
40
19.4
68.6
121.7
MIMIC-IV
20
20
20
6.4
73.1
177.1
Missing values in the risk factor vectors imputed using the most recent non-missing value from prior encounters (i.e., last-observation carried forward). This approach preserves temporal consistency and aligns with clinical practice, where outdated test results often referenced until updated.
In addition, under guidance from clinical collaborators, several continuous-valued physiological indicators were discretized into clinically meaningful categories. Specifically:
GFR values were discretized into GFR_NORM (2 90), GFR.Decrease.Slight (60 < GFR 90), and GFR Decreaace Severe (< 60).
HDL values were mapped to HDL.Good (2 60), Normal (40 < HDL < 60), and HDL_Bad (< 40).
Triglycerides were categorized into
Triglycerides Good (< 150),
Triglycerides LowRisk (150 TG 199), and Triglycerider Hi Highissk (2 199).
These discrete categories were embedded as input tokens for our model.
For the MIMIC-IV dataset, only structured diagnosis codes were used construct the complication marker vectors (lx) by mapping ICD-9/10 codes to the predefined marker set V. Risk factor inputs were not utilized due sparsity and inconsistency across patient records.
Hypergraph Construction. We constructed disease progression hypergraph 7. based on expert- validated trajectories of diabetes complications. Each trajectory forms hyperedge capturing high- order progression patterns among markers, serving as the structural backbone for constructing patient-specific TD-Hypergraphs in our framework.
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Published as a conference paper at ICLR 2026
D.5 IMPLEMENTATION DETAILS
We applied the same label preprocessing strategy to both datasets: for each complication marker, only its first occurrence (onset) was used. This transformed the problem from simple state classi- fication to the actual challenge, i.e., disease onset prediction - whether a new complication would occur in the next encounter. For both datasets, we randomly split patients into training, validation, and test sets using an 8:1:1 ratio.
We implemented all models using PyTorch and conducted training on cluster of 8XNVIDIA A.00 GPUs (80GB). Our TD-HNO DD-ONODE model was trained using the Adam optimizer with learning rate of le-4, weight decay of 1e-6, and batch size of I. Training was run for up to 200 epochs, with early stopping applied based on the validation loss (patience 5).
We used 228ddimessional embeddings for both discrete token inputs and continuous time position embedding. The hyperedge attention encoder consisted of 2 attention layers with 8 attention heads each, a feed-forward expansion factor 4, GELU activation, and dropout rate 0.1. Positional encodings were learnable, and token embeddings were initialized using uniform distribution.
The Neural ODE module modeled continuous-time latent dynamics over the TD-Hypergraph using fixed-step Runge-Kutee Kutta 4th-order method (RK4) with 10 solver steps. The input to the ODE consisted the hypergraph-enhante -enaanccd representations obtained from the TD-Hygergaaph Laplacian, and its outputs were decoded into complication marker probabilities via sigmoid layer.
Source code has been included in the Supplementary Material for reproducibility.
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Published as a conference paper at ICLR 2026
D.6 BASELINES DETAILS
To evaluate the effectiveness of TD-HNODE, we compare against representative baselines from several major categories:
(1) Sequential Models:
T-LSTM Baytas et al. (2017): time-aware LSTM variant designed handle irregularly sampled patient records.
2024b): Extends the Transformer architecture by redefining self-attention to operate over time-evolving latent trajectories.
ContiFormer Chen et al.
(2) Temporal Graph Neural Networks:
MegaCRN
Jiang et al. (023): A discrete-time GNN model that combines graph con- volution with gated recurrent units (GRU), operating on temporal graphs represented as discrete-time snapshots.
TGNE et al. (2024): A continuous-time GNN that constructs event-based structural messages to model evolving temporal graphs in fine-gr one-grained manner.
(3) Temporal Hypergraph Neural Networks:
DHSL Wang et al. (2024)): Combines hypergraph convolution network with GRU to jointly model oige-order spatial correlations and temporal dependencies.
HyperTime Younis and Ahmadi (2024)) Constructs dynamic hypergraphs from time series segments and applies LSTM-enhanced hyperedge convolutions to model evolving temporal patterns.
(4) Neural ODE-based Models:
NODE Chen et al. 2018): : A foundational continuous-timg model that captures smooth temporal dynamics ODE-basee latent evolution.
CODE-RNN Coelho al. (0255): Combines Neural ODEs with recurrent networks to model temporal dynamics time-series data.
As for DHSI. and HyperTime, both adopt discrete-time hypergraph neural network: They divide continuous time domain into discrete time intervals (different snapshots). Within each time inter- val, they construct static hypergraph snapshot and use hypergraph convolution to extract spatial structure. Then, recurrent module (GRU/LSTM) is applied across the snapshots to propagate fea- tures over time. This design may encounter difficulties in modeling the subtle temporal progression patterns on irregular time data (as in our case).
To ensure fairness, we customized each method to our task setting. Specifically, for models without structural inputs, i.e., Categories (1) and (4), concatenate the risk factor vector and marker status vector y as inputs. For graph-based models, i.e., Categories (2), we break the TD-Hypergraph H into standard graph with pairwise edges. For models that assume discrete time, i.e., MegaCRN, DHSL, HyperTime, we segment the temporal graph or hypergraph into fixed-length snapshots by treating each patient encounter a separate timestamp. That is, we construct static graph (or hypergraph) for each encounter and stack them sequentially discrete time steps. For graph-based models that assume continuous time, iee, TGNE, we treat each encounter as an individual event, ordered by its timestamp. We convert the temporally detailed hypergraph into sequence of times- tamped edges.
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Published as a conference paper at ICLR 2026
D.7 ADDITIONAL EXPERIMENTS ON CHRONIC DISEASE
To further evaluate the generalizability of our framework, we incorporated an additional chronic disease (cardiovascular disease) using the publicly available MIMIC-IV dataset JJonnson et al. 2003). ICD-9/10 codes were mapped to five clinically recognized markers: Hypertension, Atrial Fibrillation, Heart Failure, Cerebrovascular Disease Stroke, and Myocardial Infarction. Based on prior clinical studies Dzeshka et al. (2015): Bonow et al. (2011). we defined three representative progression pathways:
Hypertension - Atrial Fibrillation - Heart Failure
Hypertension Myocardial Infarction + Heart Failure
Hypertension Cerebrovascular Disease Stroke
This preprocessing resulted in 1,665 patients (train//allddaiionnesss 1,332/166/167), with dataset statistics summarized in Table
Table 7: Statistics of the cardiovascular disease dataset.
Number of Encounters
Time Span (Months)
Dataset
Min
Avg
Max
Min
Avg
Max
Cardiovascular Disease
9
20
4.2
53.3
94.2
We compared TD-HNODE with baseline methods, and the results are presented in Table
Table 8: Performance comparison on the cardiovascular disease dataset.
Model
Accuracy
Precision
Recall
Fl-score
T-LSTM
0.701
0.126
0.613
0.179
ContiFormer
0.796
0.189
0.765
0.266
MegaCRN
0.744
0.167
0.759
0.233
TGNE
0.761
0.175
0.801
0.247
DHSL
0.740
0.150
0.717
0.232
HyperTime
0.752
0.160
0.734
0.231
NODE
0.786
0.152
0.689
0.227
CODE-RNN 0.779
0.166
0.699
0.190
TD-HNODE 0.804
0.193
0.818
0.291
As shown in Table 8 TD-HNODE consistently outperforms all baselines on the cardiovascular
discase dataset, particularly in recall and F1-score. These findings demonstrate that framework effectively captures the progression of chronic diseases beyond diabetes and highlight its potential for generalization when expert-defined progression pathways available.
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Reviews:
[{"review_id": "UxQW0LdhGt", "reviewer": "Reviewer_WqcR", "summary": "In this work, the authors propose TD-HNODE that models disease progression along clinically recognized trajectories by constructing a temporally detailed hypergraph and capturing continuous-time progression dynamics through a neural ODE framework.", "questions": "See the weakness.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "1. This provides a novel modeling method for EHR.\n\n2. The figure is clearly diagrammed, and the notation table enhances the readability.\n\n3. Both open-source and closed-source datasets are evaluated, validating its practice.", "weaknesses": "1. It is limited to evaluating the proposed method on only one category of EHR, i.e., type 2 diabetes. Other medical scenarios, e.g., Alzheimer’s disease, Parkinson’s disease, and chronic kidney disease (CKD), mentioned by the authors, are ignored. \n\n2. The evaluation lacks soundness. Firstly, it is encouraged to involve the doctors' diagnosis by comparison. Secondly, the ODE steps need to extend to evaluate the robustness of TD-HNODE.\n\n3. The writing is quite informal, e.g., we use the terms ‘hyperedge’, ‘pathway’, and ‘trajectory’ interchangeably.\n\n4. The crucial baseline that models graph ODE, is ignored, e.g., HOPE[1].\n\n5. The induction of equation 11 is absent. Why can an ODE capture this dynamic Laplacian matrix? \n\n6. The related work is not explicitly illustrated.\n\n[1] Luo, Xiao, et al. \"Hope: High-order graph ode for modeling interacting dynamics.\" International conference on machine learning. PMLR, 2023.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "8ZDBxb5ChO", "reviewer": "Reviewer_rGrB", "summary": "This paper addresses the prediction of disease progression from patient encounter data, incorporating risk factors such as medications, laboratory test results, and vital signs. The authors propose TD-HNODE, which combines a neural ODE and a hypergraph neural network, where Learnable TD-Hypergraph Laplacian plays a key role to make the model more data-driven and adaptable to patient-specific disease trajectories while maintaining its clinically verified nature. The proposed method was evaluated on two real-world EHR datasets and consistently outperformed baselines.", "questions": "Nothing.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 4, "strengths": "-- Learnable TD-Hypergraph Laplacian is a reasonable enhancement for the combination of neural ODE and hypergraph neural network, where Attention-based Indicence Matrix adjusts the degree of attributions of v in e flexibly according to the context, and Learnable Hyperedge Weight Matrix captures data-driven similarities between trajectories.\n\n-- Experimental results on multiple datasets demonstrated the effectiveness of the proposed method.\n\n-- The case study is interesting and practically important.", "weaknesses": "-- Clarity issues:\n* In l.100, what is k?\n* In l.147, The authors mentioned \"we use the terms ‘hyperedge’, ‘pathway’, and ‘trajectory’ interchangeably\", but this makes descriptions confusing. For example, p_j and e_j look the same, so we should use only one of them consistently throughout the paper.\n* In Eq.2, LHS is better to be f(t,S(t),x(t);\\Theta) not dS(t)/dt.\n* In Eq.3, I is not defined.\n* In the descriptions starting from l.242, e is used like an index of edge, but it was j until then. The index for the edge should be j, and the edge itself is denoted as e for consistency.\n* In l.265, the temporally detailed hyperedge should have u on superscript.\n* In l.269, e may not be the index for O and F, maybe. For indexing, only j is enough.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "sLKFQvbKfE", "reviewer": "Reviewer_CKa4", "summary": "TD-HNODE is a neat and technically sound synthesis -- temporally detailed hypergraph structure inside a neural ODE which is complemented with empirical improvements and ablations that attribute gains to both the attention based incidence and learnable trajectory weights.\n\nOverall, I find the paper easy to read, well-written, and reasonably reproducible with the code provided as supplementary material. Further, it addresses a practical and relevant clinical modelling gap. The main caveats are clinical validity of the \"irreversible\" and pathway assumptions, narrow evaluation metrics for deployment, and baseline coverage -- which I would like the reviewers to clarify further.", "questions": "The questions that I have for the authors are described in the weaknesses of the paper that I mentioned, further I would like the authors to clarify the following,\n\n- Tables report mean/std but not CIs/significance tests, and there’s no cross-site validation - why is that?\n- There is also no analysis of shift when varying hypergraph definitions, and limited reporting on class prevalence which is key for clinical claims?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "Overall, the paper is well-motivated and tackles a problem of high relevance; namely addressing continuous-time disease progression with irregular visits and aligns modeling with clinically recognized pathways.\n\nSummarising the positives below,\n\n- Methodological novelty with clear mechanics. TD-HNODE combines a Neural ODE with a temporally detailed hypergraph: an attention-based, time-aware incidence matrix and a learnable hyperedge weight matrix to form a TD-Hypergraph Laplacian that drives ODE dynamics. This is a neat way to infuse high-order, pathway-level structure into continious modelling.\n- Good experimentation including two real-world EHR datasets with patient-wise splits; additional cardiovascular disease experiments suggest some generality beyond diabetes which could further enhance the method applicability.\n- Clear gains vs strong baselines + informative ablations. TD-HNODE tops Accuracy/Recall/F1 across T-LSTM, ContiFormer, TGNE, etc.\n- Code is supplied in the supplementary material which greatly enhances reproducibility and dissemination of the work.", "weaknesses": "On the weaknesses for the paper can be summarised below,\n\n- Labeling/trajectory assumptions may be clinically brittle. All markers are treated as irreversible first-occurrence events; including lab states like \"HbA1c High/Low\", \"Poor Lipid/BP\". This monotonicity simplifies modeling but risks mischaracterizing fluctuating conditions and inflating cumulative positives which can inflate results.\n- Metric story is narrow for a clinical setting. Results emphasize Recall/Accuracy and macro F1; there’s no AUROC/AUPRC, calibration, decision-curve or time-to-event analysis. This is rather important in this setting as there are class imbalances and early-warning goals. Overall precision remains relatively low (e.g., 31.8% on MIMIC-IV), so false-positive indidence remains unclear.\n- Baseline customization may under-serve competitors. Graph baselines are forced into pairwise graphs (breaking hyperedges), while discrete-time hypergraph baselines are snapshot-based. That’s fair to compare paradigms, but the paper should also include at least one continuous-time hypergraph variant.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ChESFqQs4e", "reviewer": "Reviewer_wNfJ", "summary": "This paper introduces TD-HNODE, a disease progression model that integrates clinical knowledge into a temporally detailed hypergraph combined with Neural ODE. Each node represents a disease complication marker, and each hyperedge is a predefined clinically validated progression trajectory. For disease modeling, the authors propose a time-adaptive Laplacian that governs continuous-time diffusion of latent marker states, comprising an attention-based incidence matrix for patient-specific, time-aware weighting of markers and a learnable hyperedge weight matrix. Experiments on two EHR datasets (University Hospital and MIMIC-IV) show that TD-HNODE improves accuracy, recall, and F1 compared with strong baselines (T-LSTM, ContiFormer, TGNE, HyperTime, CODE-RNN). Ablations support the contributions of both adaptive incidence $H_p$ and learnable weights $W_p$, and a case study suggests that the model’s latent embeddings reveal sub-phenotypes of diabetic progression.", "questions": "1. Please detail how predefined trajectories were constructed (clinical source, inter-rater agreement, # of trajectories, examples). \n2. Since hyperedges are set-valued, how is order encoded beyond attention? \n3. Why not construct trajectories from actual temporally detailed sequences (e.g., frequent pattern mining, partial order mining), then regularize by clinical priors? \n4. Classical hypergraph Laplacians rely on diagonal W to ensure symmetry and PSD. With dense $W_p$, $\\tilde L$ may not be symmetric/PSD unless extra constraints hold. Non-PSD can destabilize diffusion and the ODE. Please prove conditions under which $\\tilde L(t)$ is symmetric/PSD with dense $W_p$. \n5. Please specify the exact initialization: $S(t_1)=\\mathrm{Enc}([x(t_1), y(t_1)])$", "limitations": "", "rating": 2, "confidence": 4, "soundness": 2, "presentation": 1, "contribution": 2, "strengths": "1. The topic is important. Continuous-time modeling of chronic disease trajectories is an important and emerging topic. \n2. Interpretability might be good, as the hyperedges align with known clinical pathways. This might serve as a foundation for explanation and clinical validation. \n3. Consistent improvements on two EHR datasets, particularly in recall (clinically critical for early detection).", "weaknesses": "1. Over-complex and arguably unnatural construction.\nWhile the idea of embedding medical knowledge into continuous-time dynamics is valuable, the resulting architecture feels heavily engineered. TD-HNODE stacks many modeling layers—curated trajectories $\\rightarrow$ hypergraph $\\rightarrow$  attention-based incidence $\\rightarrow$  dense inter-trajectory weighting $\\rightarrow$  ODE integration—each adding parameters without clear generative justification. It is difficult to discern whether the model captures meaningful structure or merely benefits from large capacity.\n2. Everything is learnable, risking loss of inductive bias. Almost all structural components ($H_p$, $W_p$, time encodings) are trainable.\nThis undermines the “knowledge-infused” motivation: the learned Laplacian may diverge from curated pathways, reducing interpretability.\nRegularization toward clinical priors or partial parameter freezing would help maintain domain grounding.\n3. Bias compounding across multiple submodules.\nThe architecture effectively stacks a model on top of another (attention $\\rightarrow$ self-attention $\\rightarrow$ pooling $\\rightarrow$ ODE $\\rightarrow$ decoder). Each stage may introduce its own bias, and the composition could amplify rather than mitigate error.\nIt remains unclear which layer drives performance versus redundancy.\n4. Questionable scalability of the hypergraph construction.\nThe temporally detailed hypergraph may require combining all observed markers across trajectories, potentially leading to a combinatorial explosion of hyperedges. The paper does not quantify computational or memory costs beyond brief remarks in the appendix.\nRecomputing dense, time-varying $\\tilde{L}(t)$ could be infeasible for large EHRs.\n5. Clarity and naturalness of formulation. The path from clinical intuition to mathematical formulation is hard to follow. Although mathematically consistent, the paper could better motivate why this sequence of modeling choices is natural or necessary.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "CHrSkKnajK", "reviewer": "Reviewer_F3q3", "summary": "This study introduces a novel method that combines Neural ODE and GNN–based for interpretable trajectory generation.\nThis work uses hypergraph neural network to capture relational dependencies among variables and uses neural ODE to learn continuous dynamics.\nThe authors further provide theoretical support ensuring representational validity.\nExperiments demonstrate that the method achieves accurate trajectory reconstruction and offers interpretable latent dynamics.", "questions": "Extending the weakness 2. \n\n1.  While the motivations mentions subpheontypes, did the author try to analyze it? whether it is connected the different progression speed in Fig.4. \n\n2. This work focuses on T2D and selects features to analyze it. Did the author analyze the correlation or combritidy between T2D and other features? whether T2D will also contribute to other diseases (like heart failure)\n\n3. While the background mentions medication and talks about \"timely treatment\", it does not have any thing about the interactions between disease progression and medications? we should see medication helps the disease recovery? \n\n4. Some symptoms are irreversible. I am wondering whether the medictions can help stabalize it? It is interesting that the authors mention irreversible symptoms but it does not make analysis in the result section.", "limitations": "", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "1. It combines Neural ODEs and GNNs to effectively capture both temporal and spatial dependencies. \n\n2. It improves over the traditional fixed hyperedge weight matrix by using a learnable hyperedge weight matrix. \n\n3. By modeling continuous dynamics and graph relations, the model offers interpretable latent representations and visualizable trajectories that help in analyzing its validity. \n\n4. The experimental section shows superior reconstruction and forecasting accuracy compared with baseline methods.\n\n5. It provides visualization for the generated trajectories.", "weaknesses": "1. This paper lacks a comprehensive investigation of Transformer based model for irregularly-sampled event sequence.\n\nFor Transformer models, it should include but not limited: \n\n[1] Learning the natural history of human disease with generative transformers. Nature 2025. \n\nFor linear Transformer, it can be considered as discreted ODE, for instance:\n\n[2] TrajGPT: Irregular Time-Series Representation Learning of Health Trajectory. IEEE J-BHI 2025. \n\nI just list few recent works. You could find more related works which should be included in the literature review.\n\n2. Although this paper claims that it can generate interpretable trajectory, it has limited data analysis about the generated trajectories. It should include more case studies about how a patient's health progress over time. It could also include population-level analysis about the subphenotypes or combritidy. \n\n3. It lacks interpretation and visualization of the learned token embedding (like HP, AF). We do not know whether model learns meaingful embedding or not in the graph-based model.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)