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Neural structure learning with stochastic differential equations
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Proposes SCOTCH, a Bayesian structure learner that combines latent neural SDEs with variational inference over graphs, with identifiability and consistency results, and outperforms baselines on synthetic and real time-series structure learning under regular and irregular sampling.
Problem
Learning the causal/structural graph among variables from temporal observations when the underlying process is continuous-time and observations may be irregularly sampled; existing discrete-time structure learners misrepresent continuous dynamics and handle irregular sampling poorly.
Approach
Model the latent dynamics as a graph-parameterised neural Itô SDE (with diagonal diffusion) plus observational noise, place a Bernoulli variational posterior over graphs, and optimise a variational ELBO over latent path posteriors, yielding identifiability and consistency guarantees in continuous time.
Key results
- On Lorenz-96 with 30% missingness SCOTCH reaches AUROC 0.7279 vs 0.6788 for NGM and 0.5714 for Rhino.
- On Glycolysis (full data) SCOTCH achieves AUROC 0.7113 vs 0.5953 for NGM.
- On DREAM3 100-dimensional networks SCOTCH averages AUROC 0.677 vs 0.636 for Rhino and 0.580 for NGM.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use SCOTCH instead of NGM | Lorenz-96 structure learning · Lorenz-96 p=0.3 | AUROC: 0.6788$\pm$0.009 -> 0.7279$\pm$0.017 | Helped |
| Use SCOTCH instead of CUTS | Lorenz-96 structure learning · Lorenz-96 p=0.6 | AUROC: 0.6418$\pm$0.012 -> 0.6453$\pm$0.014 | No clear effect |
| Use SCOTCH instead of NGM | Glycolysis structure learning · Glycolysis Full | AUROC: 0.5953$\pm$0.018 -> 0.7113$\pm$0.012 | Helped |
| Use SCOTCH instead of Rhino | DREAM3 structure learning · DREAM3 Mean | AUROC: 0.636$\pm$0.022 -> 0.677$\pm$ 0.026 | Helped |
| Use SCOTCH instead of Rhino+NoInst | Netsim structure learning · Netsim Full | AUROC: 0.95 $\pm$0.001 -> 0.95$\pm$ 0.006 | No clear effect |
| Use SCOTCH instead of Rhino+NoInst | Netsim structure learning · Netsim p=0.1 | AUROC: 0.93$\pm$ 0.005 -> 0.91$\pm$0.007 | Hurt |
| Use SCOTCH instead of Rhino+NoInst | Netsim structure learning · Netsim p=0.2 | AUROC: 0.90$\pm$0.012 -> 0.89$\pm$0.007 | Hurt |
Surprising findings
- On Netsim, the Rhino-based discrete model (Rhino without instantaneous effects) performs on par with or better than SCOTCH, which the authors attribute to the data being better modelled as discrete.
- Rhino with full instantaneous effects reaches AUROC 0.99 on Netsim, above SCOTCH, which cannot model instantaneous effects.
Limitations
- Cannot handle instantaneous effects, which can arise from data aggregation.
- Computational cost scales linearly with series length since the SDE is solved over the full trajectory.
- Homogeneous drift and diffusion mean it cannot model non-stationary systems.
Experimental setup
Datasets: Lorenz-96 Glycolysis DREAM3 Netsim
Benchmarks: Lorenz-96 Glycolysis DREAM3 Netsim
Metrics: AUROC F1 TPR FDR
Baselines: VARLiNGaM PCMCI+ CUTS Rhino NGM
Builds on: NGM Rhino CUTS latent SDE
Closest papers by meaning
- SING: SDE Inference via Natural Gradients (NeurIPS 2025 · similarity 0.87)
- In-Context Learning of Stochastic Differential Equations with Foundation Inference Models (NeurIPS 2025 · similarity 0.87)
- Efficient Training of Neural Stochastic Differential Equations by Matching Finite Dimensional Distributions (ICLR 2025 · similarity 0.86)
- Neural Stochastic Flows: Solver-Free Modelling and Inference for SDE Solutions (NeurIPS 2025 · similarity 0.86)
- Variational Inference for SDEs Driven by Fractional Noise (ICLR 2024 · similarity 0.86)
- Closing the Approximation Gap in Simulation-free Latent SDEs (NeurIPS 2026 · similarity 0.86)
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