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Neural Statistical Functions

NeurIPS 2026 · Probabilistic and scientific modeling › Uncertainty and Gaussian processes · approach: Approximate inference in Bayesian neural networks · 0 citations (OpenAlex)

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Introduces neural statistical functions that learn prefix statistics over operating-condition intervals from a pretrained single-sample predictor plus scattered data, so interval means, maxima, and quantiles need only a few endpoint evaluations instead of Monte Carlo sampling.

Problem

Estimating statistics (integrals, quantiles, maxima) of physical responses over ranges of operating conditions normally requires many repeated single-condition surrogate or solver evaluations, which is slow for design and reliability analysis.

Approach

Learn a neural network for the prefix statistic (integral from the lower bound up to a condition) and train it with a first-order identity s·∂s h̃ + h̃ = ψ, so any interval statistic follows from two endpoint evaluations; quantiles and maxima are reduced to the same prefix-integral template via transformed signals (indicator for CDF, exp(βh) for log-sum-exp max).

Key results

• On 2D dynamics, neural statistical functions with two endpoint evaluations achieve relL2 0.028 at width 0.50 against analytic reference, versus 0.096 for MC K=10 and 0.016 for MC K=400.
• On Car-Crash maximum stress at 100° width, the method reaches relL2 0.335 (w/ data supervision) versus 0.322 for MC K=10 against the dense Transolver reference, which is competitive with MC while using only endpoint evaluations.
• Abstract reports up to a 100x reduction in model evaluations.

Surprising findings

• On Car-Crash maxima against the dense Transolver reference, MC with K=10 slightly beats the proposed method at several widths (e.g., 0.275 vs 0.323 at width 80°).

Limitations

• Experiments only use one-dimensional condition variables; multi-dimensional extension costs 2^d evaluations and is untested.
• Quantile and maxima quality depends on predictor bias and on smoothing hyperparameters (beta, epsilon); the crash max result is not uniformly better than MC K=10.
• Some references are themselves surrogate outputs (Transolver dense sweeps) rather than ground-truth simulations.

Experimental setup

Datasets: synthetic 2D dynamical system (2,500 samples) NASA-CRM (149 samples) Car-Crash OpenRadioss Neon simulations (280 cases)

Benchmarks: 2D dynamics interval mean NASA-CRM quantile pressure Car-Crash maximum stress

Metrics: width-binned relative L2 error number of model evaluations

Baselines: Monte Carlo sampling with single-condition predictor (K=5,10,20,50,100,200,400) Transolver dense sweeps

Builds on: Transolver Neural ODE log-sum-exp smooth maximum

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