CoNBONet: Conformalized Neuroscience-inspired Bayesian Operator Network for Reliability Analysis

Shailesh Garg; Souvik Chakraborty

stat.MLMar 23, 2026

CoNBONet: Conformalized Neuroscience-inspired Bayesian Operator Network for Reliability Analysis

Authors: Shailesh Garg, Souvik Chakraborty

Organizations: Department of Applied Mechanics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India. · Department of Applied Mechanics, Yardi School of Artificial Intelligence (YScAI), Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India.

Abstract

Time-dependent reliability analysis of nonlinear dynamical systems under stochastic excitations is a critical yet computationally demanding task. Conventional approaches, such as Monte Carlo simulation, necessitate repeated evaluations of computationally expensive numerical solvers, leading to significant computational bottlenecks. To address this challenge, we propose CoNBONet, a neuroscience-inspired surrogate model that enables fast, energy-efficient, and uncertainty-aware reliability analysis, providing a scalable alternative to techniques such as Monte Carlo simulations. CoNBONet, short for Conformalized Neuroscience-inspired Bayesian Operator Network, leverages the expressive power of deep operator networks while integrating neuroscience-inspired neuron models to achieve fast, low-power inference. Unlike traditional surrogates such as Gaussian processes, polynomial chaos expansions, or support vector regression, that may face scalability challenges for high-dimensional, time-dependent reliability problems, CoNBONet offers fast and energy-efficient inference enabled by a neuroscience-inspired network architecture, calibrated uncertainty quantification with theoretical guarantees via split conformal prediction, and strong generalization capability through an operator-learning paradigm that maps input functions to system response trajectories. Validation of the proposed CoNBONet for various nonlinear dynamical systems demonstrates that CoNBONet preserves predictive fidelity, and achieves reliable coverage of failure probabilities, making it a powerful tool for robust and scalable reliability analysis in engineering design.

Figures & tables

Figure 1: Schematic for information flow in CoNBONet.

Panel of Figure 2 — Figure 2: Ratio of energy consumed by densely connected ANN layer vs densely connected VSN layer, when run for a single Spike Time Step (STS). It is assumed that all output nodes will produce spikes.

Panel of Figure 3 — Figure 3: Ratio of energy consumed by densely connected ANN layer vs densely connected VSN layer, when run for two STSs.

Figure 4: Mean prediction from the CoNBONet compared against the ground truth with calibrated confidence intervals shown as the blue patch, for example E-I.

Layer	A1	A2
1 st DOF	14.70	18.20

Table 1: Spiking activity (%) for the two layers (A1 and A2) of branch network when using CoNBONet for example E-I.

Model	Coverage (%)			Number of time steps with coverage
	Avg.	Min.	Max.	$<95\%$	$\geq 95\%$
BONet	93.98	81.46	100.00	53	47
CoBONet	99.73	98.68	100.00	0	100
NBONet	97.56	89.00	99.98	2	98
CoNBONet	97.76	96.92	99.86	0	100

Table 2: Coverage provided by the confidence intervals generated using various network in example E-I.

Figure 5: Probability of failure at different time steps in example E-I.

Mass (kg)	Stiffness (N/m)	Damping (Ns/m)
$m_{1}=10$	$k_{1}=10000$	$c_{1}=100$
$m_{2}=10$	$k_{2}=10000$	$c_{2}=100$
$m_{3}=9$	$k_{3}=9000$	$c_{3}=90$
$m_{4}=9$	$k_{4}=9000$	$c_{4}=90$
$m_{5}=7.5$	$k_{5}=7500$	$c_{5}=75$

Table 3: System parameters for Case-II: five-DOF (5-DOF) Duffing oscillator.

Panel of Figure 6 — Figure 6: Mean prediction from the CoNBONet compared against the ground truth with calibrated confidence intervals shown as the blue patch, for example E-II.

Network	1 st DOF	2 nd DOF	3 rd DOF	4 th DOF	5 th DOF
BONet	0.0018	0.0019	0.0027	0.0009	0.0018
CoNBONet	0.0048	0.0060	0.0022	0.0044	0.0051

Table 4: NMSE for each degree of freedom obtained when comparing network predictions against the ground truth in E-II.

Layer	1 st DOF	2 nd DOF	3 rd DOF	4 th DOF	5 th DOF
A1	11.5	12.5	14.0	15.0	16.1
A2	17.5	14.0	16.2	22.5	16.4

Table 5: Spiking activity (%) for the two layers (A1 and A2) of the branch network in CoNBONet (Example E-II).

		Coverage (%)			Number of time steps with coverage
DOF	Model	AvgCov	MinCov	MaxCov	$<95\%$ / $\geq 95$ (%)
1 st	BONet	95.30	83.72	99.94	29 / 71
	CoBONet	99.23	98.18	99.96	0 / 100
	NBONet	98.37	97.54	100.00	0 / 100
	CoNBONet	98.52	97.24	99.98	0 / 100
2 nd	BONet	92.97	75.78	100.00	59 / 41

Table 6: Coverage provided by the confidence intervals generated using various network in example E-II.

Panel of Figure 7 — Figure 7: PoF obtained using FTTF analysis at various DOFs in E-II. The PoF is computed corresponding to different threshold vales at each DOF.

	DOF 0	DOF 1	DOF 2	DOF 3	DOF 4
Threshold	0.25	0.48	0.67	0.82	0.91
True PoF	0.633	0.617	0.607	0.599	0.596
$\underset{\text{Bounds}}{\text{Mean PoF}}$	$\underset{[0.584,\,0.736]}{0.628}$	$\underset{[0.581,\,0.694]}{0.614}$	$\underset{[0.569,\,0.685]}{0.603}$	$\underset{[0.563,\,0.718]}{0.596}$	$\underset{[0.557,\,0.694]}{0.593}$

Table 7: PoF obtained using FTTF analysis at various DOFs in E-II. The PoF is computed at each DOF, correspondin to a single set of threshold values.

Panel of Figure 8 — Figure 8: Mean prediction from the CoNBONet compared against the ground truth with calibrated confidence intervals shown as the blue patch, for example E-III.

Network	5 th DOF	15 th DOF	35 th DOF	65 th DOF	75 th DOF
BONet	0.0083	0.0027	0.0021	0.0011	0.0004
CoNBONet	0.0088	0.0072	0.0125	0.1101	0.0132

Table 8: NMSE for each degree of freedom obtained when comparing network predictions against the ground truth in E-III.

Layer	5 th DOF	15 th DOF	35 th DOF	65 th DOF	75 th DOF
A1	12.4	23.0	09.5	08.2	11.4
A2	31.3	15.0	24.3	09.4	08.0

Table 9: Spiking activity (%) for the two layers (A1 and A2) of the branch network in CoNBONet (Example E-III).

DOF	Model	AvgCov	MinCov	MaxCov	$<95$ / $\geq 95$ (%)
5 th	BONet	85.91	76.36	100.00	86 / 14.00
	CoBONet	98.66	97.54	100.00	0 / 100.00
	NBONet	82.72	54.00	99.80	88 / 12.00
	CoNBONet	99.02	97.62	99.72	0 / 100.00
15 th	BONet	89.97	81.18	99.96	73 / 27.00
	CoBONet	98.95	98.12	99.96	0 / 100.00

Table 10: Coverage provided by the confidence intervals generated using various network in example E-III.

Figure 9: PoF obtained using FTTF analysis at 76 th DOF in E-III. The PoF is computed corresponding to different threshold vales.

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Conformal Prediction for Neural Operators: Distribution-Free Uncertainty Quantification in Physics Simulation

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Neural operators such as the Fourier Neural Operator (FNO) have emerged as powerful surrogates for solving partial differential equations (PDEs), achieving speedups of several orders of magnitude over traditional numerical solvers. However, deploying these models in safety-critical engineering applications -- such as thermal management of electronic components and battery systems -- requires not only accurate point predictions but also rigorous uncertainty guarantees. Existing uncertainty quantification (UQ) methods for neural operators, including Monte Carlo Dropout and Deep Ensembles, provide only relative uncertainty estimates without formal coverage guarantees. In this work, we propose the first application of split conformal prediction to neural operator-based physics simulation, providing distribution-free prediction intervals with finite-sample coverage guarantees. We further introduce a normalized conformal prediction scheme that leverages MC Dropout uncertainty to produce adaptive-width intervals, yielding tighter intervals in regions of low uncertainty and wider intervals where the model is less certain. Full-scale experiments (33.7M parameters, 800 training samples, 5 ensemble members, NVIDIA V100) on steady-state heat conduction benchmarks demonstrate that our method achieves 89.1% empirical coverage at the target level of alpha=0.1, while producing spatially adaptive prediction intervals that reflect the underlying physical uncertainty structure. We also provide an uncertainty decomposition framework that separates epistemic uncertainty (68% of total) from aleatoric uncertainty (32% of total), offering actionable guidance for data collection and model improvement. Our method is implemented in an open-source platform with REST API endpoints and interactive 3D visualization.

Abstract

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