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A Scalable Nonparametric Continuous-Time Survival Model through Numerical Quadrature

NeurIPS 2026 · Probabilistic and scientific modeling › Dynamical systems · approach: Latent dynamics models · 0 citations (OpenAlex)

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Proposes QSurv, a nonparametric continuous-time survival model that parameterizes the log-hazard with a neural network, evaluates the cumulative hazard via Gauss-Legendre quadrature for end-to-end training, and conditions backbones on time with Time-LoRA.

Problem

Training flexible continuous-time survival models that directly parameterize the instantaneous hazard is hard because the cumulative hazard integral in the likelihood is intractable, and existing options rely on discretization, proportional hazards, parametric assumptions, or costly ODE solvers.

Approach

Model the log-hazard with a network and approximate the cumulative hazard by K-point Gauss-Legendre quadrature inside the NLL (differentiable, O(K) evaluations), plus a Time-LoRA head that modulates penultimate-layer weights by time so backbone features can be cached across quadrature nodes.

Key results

• On simulations, QSurv's instantaneous hazard L1 error is competitive with SODEN (e.g., Exponential 0.0352 vs 0.0330) while training ~6 s vs ~60 s on average.
• On C4KC-KiTS QSurv achieves the best C_td at all horizons (e.g., 0.6381 full horizon vs next best 0.5808 CoxTime).
• On tabular datasets QSurv is generally competitive, often matching best baselines, with smaller performance gaps than on imaging.

Surprising findings

• Direct concatenation of time outperforms Time-LoRA on several simulated distributions (Weibull, Gamma, Log-normal), per the ablation.
• On COVID-19-NY, QSurv does not dominate on IBS/IBLL, where DeSurv and SODEN are better.

Limitations

• Advantages over baselines are often within standard deviations on tabular data.
• Time-LoRA restricted to penultimate layer; applying to intermediate layers would require full backbone passes per node.
• Performance may degrade under severe censoring, distribution shift, or highly irregular hazards (acknowledged in conclusion).

Experimental setup

Datasets: GBSG METABRIC NWTCO FRAMINGHAM FLCHAIN SUPPORT2 COVID-19-NY C4KC-KiTS BraTS

Benchmarks: GBSG METABRIC NWTCO FRAMINGHAM FLCHAIN SUPPORT2 COVID-19-NY C4KC-KiTS BraTS

Metrics: C_td Integrated Brier Score IBLL D-calibration L1 error of hazard, cumulative hazard, survival

Baselines: CoxCC CoxTime DeepHit NnetSurv MDN DeSurv SODEN

Builds on: DeSurv SODEN CoxTime DeepHit FiLM

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