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Graph Neural Flows for Unveiling Systemic Interactions Among Irregularly Sampled Time Series
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Proposes GNeuralFlow, a graph neural flow that learns a DAG (Bayesian network / SEM-style) over irregularly sampled time series jointly with an ODE-solver-free neural flow parameterized by a GCN encoder.
Problem
Predicting and classifying dynamics of interacting multivariate time series observed at irregular times, where the interaction structure is unknown and conventional graph ODEs rely on numerical solvers or heuristic graphs.
Approach
Model interactions as a learned DAG (conditional dependencies, SEM-like) and parameterize the ODE solution directly as a neural flow (ResNet, GRU, or coupling form) whose GCN encoder uses the DAG, trained with an augmented Lagrangian enforcing acyclicity; this avoids ODE solver calls.
Key results
- On synthetic 5-node systems, GNeuralFlow (ResNet) reduces forecast MSE to 3.95e-4 on Sink versus 8.41e-4 for the ResNet neural flow and 10.6e-4 for neural ODE, and to 2.32e-3 on Triangle versus 4.01e-3 for ResNet flow.
- On Activity and Physionet smoothing classification, GNeuralFlow (GRU) reaches 0.812 AUC on Physionet (best) and GNeuralFlow (Coupling) 0.808 accuracy on Activity (best), outperforming the corresponding non-graph neural flows.
- On MIMIC-IV filtering, GNeuralFlow (GRU) achieves MSE 0.335 and NLL 0.606, the best among compared methods, improving over GRU neural flow (0.364 MSE).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add a learned DAG graph encoder to a neural flow | Neural flow (ResNet) · Sink, Triangle, Sawtooth, and Square (5-node synthetic) | MSE: 8.41 ± 0.05 -> 3.95 ± 0.32; 4.01 ± 0.52 -> 2.32 ± 0.11; 4.73 ± 0.06 -> 3.84 ± 0.06; 9.61 ± 0.02 -> 8.24 ± 0.64 | Helped |
| Add a learned DAG graph encoder to a neural flow | Neural flow (GRU) · MIMIC-IV filtering | MSE: 0.364 ± 0.008 -> 0.335 ± 0.003 | Helped |
| Add a learned DAG graph encoder to a neural flow | Neural flow (GRU) · MIMIC-IV filtering | NLL: 0.734 ± 0.054 -> 0.606 ± 0.001 | Helped |
| Use GRU flow instead of ResNet flow (ablation) | GNeuralFlow · Sink, Triangle, Sawtooth, and Square (5-node synthetic) | MSE: 3.95 ± 0.32 -> 6.83 ± 0.23; 2.32 ± 0.11 -> 5.41 ± 0.23; 3.84 ± 0.06 -> 5.11 ± 0.13; 8.24 ± 0.64 -> 9.14 ± 0.61 | Mixed |
| Use coupling flow instead of ResNet flow (ablation) | GNeuralFlow · Sink, Triangle, Sawtooth, and Square (5-node synthetic) | MSE: 3.95 ± 0.32 -> 4.45 ± 0.51; 2.32 ± 0.11 -> 3.21 ± 0.34; 3.84 ± 0.06 -> 4.25 ± 0.09; 8.24 ± 0.64 -> 8.33 ± 0.23 | Mixed |
| Use GRU flow instead of ResNet flow (ablation) | GNeuralFlow · Activity smoothing classification | Accuracy: 0.786 ± 0.009 -> 0.804 ± 0.003 | Helped |
| Use coupling flow instead of GRU flow (ablation) | GNeuralFlow · Activity smoothing reconstruction | MSE: 5.169 ± 0.05 -> 5.431 ± 0.10 | Hurt |
| Use ResNet flow instead of GRU flow (ablation) | GNeuralFlow · MIMIC-IV filtering forecast | MSE: 0.335 ± 0.003 -> 0.356 ± 0.0007 | Hurt |
| Use the ground-truth graph instead of learning the graph | GNeuralFlow · Synthetic systems | MSE | No clear effect |
| Perturb the ground-truth graph initialization more strongly | GNeuralFlow · Synthetic systems (20 nodes) | MSE | No clear effect |
Surprising findings
- Learned graphs can differ substantially from the ground truth (TPR falls and SHD rises as initialization noise grows) yet forecast MSE stays flat, and the ground-truth graph sometimes gives slightly higher MSE.
- Identifiability conditions for the DAG from the ODE model were not established; the authors suspect they may be unrealistically restrictive.
Limitations
- Parameters of the adjacency matrix grow quadratically with number of series, and DAG constraint evaluation is costly (cubic), limiting scalability to a few hundred nodes.
- No identifiability conditions established for recovering the DAG from the ODE data-generating model; learned graph may not match ground truth.
- Invertibility of GRU and ResNet flows depends on contractive components, which is only approximately enforced via regularization in practice.
Experimental setup
Datasets: Synthetic interacting systems (Sink, Triangle, Sawtooth, Square) Activity Physionet MuJoCo MIMIC-IV
Benchmarks: Synthetic systems (Sink, Triangle, Sawtooth, Square) Activity Physionet MuJoCo MIMIC-IV
Metrics: MSE accuracy AUC NLL TPR FDR FPR SHD
Baselines: Neural ODE Neural flow (ResNet) Neural flow (GRU) Neural flow (Coupling) GRU-D GDE LG-ODE CG-ODE CF-GODE NRI dNRI ODE-RNN
Builds on: neural ODE neural flows NOTEARS ODE-RNN GRU-ODE-Bayes
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