Direct Interval Propagation Methods using Neural-Network Surrogates for Uncertainty Quantification in Physical Systems Surrogate Model

Ghifari Adam Faza; Jolan Wauters; Fabio Cuzzolin; Hans Hallez; David Moens

cs.LGMar 22, 2026

Direct Interval Propagation Methods using Neural-Network Surrogates for Uncertainty Quantification in Physical Systems Surrogate Model

Authors: Ghifari Adam Faza, Jolan Wauters, Fabio Cuzzolin, Hans Hallez, David Moens

Organizations: LMSD, Department of Mechanical Engineering, KU Leuven, Heverlee, 3001, Belgium · RAM, Department of Mechanical Engineering, KU Leuven, Brugge, 8200, Belgium · Visual Artificial Intelligence Laboratory, Oxford Brookes University, Oxford, UK · DistriNet, Department of Computer Science, KU Leuven, Brugge, 8200, Belgium

Abstract

In engineering, uncertainty propagation aims to characterise system outputs under uncertain inputs. For interval uncertainty, the goal is to determine output bounds given interval-valued inputs, which is critical for robust design optimisation and reliability analysis. However, standard interval propagation relies on solving optimisation problems that become computationally expensive for complex systems. Surrogate models alleviate this cost but typically replace only the evaluator within the optimisation loop, still requiring many inference calls. To overcome this limitation, we reformulate interval propagation as an interval-valued regression problem that directly predicts output bounds. We present a comprehensive study of neural network-based surrogate models, including multilayer perceptrons (MLPs) and deep operator networks (DeepONet), for this task. Three approaches are investigated: (i) naive interval propagation through standard architectures, (ii) bound propagation methods such as Interval Bound Propagation (IBP) and CROWN, and (iii) interval neural networks (INNs) with interval weights. Results show that these methods significantly improve computational efficiency over traditional optimisation-based approaches while maintaining accurate interval estimates. We further discuss practical limitations and open challenges in applying interval-based propagation methods.

Figures & tables

Figure 1: Regularised artificial neural network (RANN) architecture for naive direct interval propagation.

Panel of Figure 2 — Figure 2: Architecture illustration for (a) Standard DeepONet architecture. (b) Modified DeepONet architecture with interval-valued branch net. The standard architecture use standard feedforward neural network (FNN) on both branches, meanwhile the modified network replance FNN with the Naive interval architecture.

Figure 3: Interval neural network (INN) with DeepONet architecture.

Figure 4: Ideal interval dataset for a one-dimensional regression problem. Single rectangle represent a single pair of interval-valued

x

and

y

$n_{\text{train}}$	Metric	Naive	IBP	CROWN	INN	Mid-CROWN	Mid-IBP	Opt-Prop
10	$\text{RMSE}_{L}$	0.076 $\pm$ 0.047	0.079 $\pm$ 0.031	0.107 $\pm$ 0.067	0.085 $\pm$ 0.051	0.049 $\pm$ 0.017	0.323 $\pm$ 0.086	0.053 $\pm$ 0.010
10	$\text{RMSE}_{U}$	0.136 $\pm$ 0.102	0.087 $\pm$ 0.057	0.109 $\pm$ 0.057	0.125 $\pm$ 0.177	0.059 $\pm$ 0.022	0.332 $\pm$ 0.113	0.063 $\pm$ 0.027
25	$\text{RMSE}_{L}$	0.025 $\pm$ 0.008	0.035 $\pm$ 0.006	0.066 $\pm$ 0.030	0.035 $\pm$ 0.030	0.040 $\pm$ 0.002	0.291 $\pm$ 0.084	0.054 $\pm$ 0.006
25	$\text{RMSE}_{U}$	0.038 $\pm$ 0.027	0.033 $\pm$ 0.006	0.059 $\pm$ 0.013	0.037 $\pm$ 0.035	0.041 $\pm$ 0.002	0.295 $\pm$ 0.083	0.045 $\pm$ 0.004
50	$\text{RMSE}_{L}$	0.023 $\pm$ 0.005	0.029 $\pm$ 0.004	0.033 $\pm$ 0.005	0.019 $\pm$ 0.004	0.040 $\pm$ 0.004	0.297 $\pm$ 0.044	0.043 $\pm$ 0.007
50	$\text{RMSE}_{U}$	0.026 $\pm$ 0.006	0.027 $\pm$ 0.002	0.049 $\pm$ 0.009	0.019 $\pm$ 0.003	0.040 $\pm$ 0.004	0.299 $\pm$ 0.045	0.053 $\pm$ 0.006

Table 1: Ideal 1-dimensional regression RMSE comparison across methods and training set sizes.

$n_{\text{train}}$	Metric	Naive	IBP	CROWN	INN	Mid-CROWN	Mid-IBP	Opt-Prop
10	PINAW	$0.999\pm 0.241$	$1.069\pm 0.198$	$1.064\pm 0.196$	$1.186\pm 0.756$	$0.620\pm 0.057$	$3.316\pm 0.215$	$0.477\pm 0.013$
	PICP	$0.742\pm 0.117$	$0.758\pm 0.060$	$0.657\pm 0.091$	$0.758\pm 0.144$	$0.558\pm 0.040$	$0.982\pm 0.012$	$0.431\pm 0.057$
	CWC	$5.287\pm 3.442$	$4.774\pm 1.349$	$7.571\pm 3.560$	$5.241\pm 2.532$	$6.395\pm 1.366$	$6.942\pm 0.395$	$9.066\pm 3.031$
25	PINAW	$1.060\pm 0.082$	$1.000\pm 0.043$	$0.932\pm 0.142$	$1.074\pm 0.110$	$0.567\pm 0.021$	$2.843\pm 0.949$	$0.475\pm 0.011$
	PICP	$0.889\pm 0.045$	$0.842\pm 0.023$	$0.706\pm 0.032$	$0.886\pm 0.084$	$0.553\pm 0.017$	$0.991\pm 0.006$	$0.465\pm 0.008$
	CWC	$2.981\pm 0.689$	$3.214\pm 0.201$	$4.946\pm 0.468$	$3.280\pm 1.768$	$5.862\pm 0.254$	$5.792\pm 1.848$	$7.385\pm 0.158$

Table 2: Ideal 1-dimensional regression performance comparison across methods and training set sizes on three different metrics: predicted interval normalized average width (PINAW), predictied interval coverage probability (PICP), and the coverage width-based criterion (CWC).

$n_{\text{train}}$	Process	Naive	IBP	CROWN	INN	Mid-CROWN	Mid-IBP	Opt-Prop
10	Training	16.504 $\pm$ 0.594	3.433 $\pm$ 0.105	24.448 $\pm$ 0.897	67.159 $\pm$ 1.609	23.467 $\pm$ 0.685	3.190 $\pm$ 0.122	12.353 $\pm$ 0.623
10	Inference	0.078 $\pm$ 0.025	0.001 $\pm$ 0.001	0.023 $\pm$ 0.001	0.213 $\pm$ 0.146	0.023 $\pm$ 0.001	0.001 $\pm$ 0.001	217.283 $\pm$ 4.548
25	Training	16.134 $\pm$ 1.021	3.394 $\pm$ 0.127	25.425 $\pm$ 0.661	67.958 $\pm$ 0.213	26.134 $\pm$ 1.349	3.316 $\pm$ 0.129	12.832 $\pm$ 0.947
25	Inference	0.075 $\pm$ 0.028	0.001 $\pm$ 0.001	0.024 $\pm$ 0.001	0.202 $\pm$ 0.156	0.023 $\pm$ 0.002	0.001 $\pm$ 0.001	218.125 $\pm$ 3.775
50	Training	30.813 $\pm$ 2.057	6.222 $\pm$ 0.222	49.373 $\pm$ 2.017	126.532 $\pm$ 7.255	51.897 $\pm$ 1.070	6.216 $\pm$ 0.032	22.484 $\pm$ 1.201
50	Inference	0.088 $\pm$ 0.066	0.001 $\pm$ 0.001	0.022 $\pm$ 0.001	0.283 $\pm$ 0.278	0.023 $\pm$ 0.001	0.001 $\pm$ 0.001	213.077 $\pm$ 7.404

Table 3: Ideal 1-dimensional regression time comparison in seconds (s). The IBP and CROWN-based methods are implemented using PyTorch and Auto_LiRPA library, meanwhile the Naive, INN, and Opt-Prop methods are implemented using TensorFlow . Therefore the training and inference time between the two groups are not directly comparable. All experiments were conducted on MacBook Pro with Apple M1 Pro chip.

Figure 5: Illustration of an ideal interval dataset for a one-dimensional PDE. The red curve denotes the input function with its interval uncertainty (dashed lines), while the blue curve denotes the corresponding PDE solution with its propagated interval uncertainty.

$n_{\text{train}}$	Metric	Naive	IBP	CROWN	INN	Mid-CROWN	Mid-IBP
50	$\text{Linex}_{L}$	$0.370\pm 0.125$	1529.564 $\pm$ 665.146	0.672 $\pm$ 0.690	8.059 $\pm$ 10.303	4.807 $\pm$ 4.204	27.859 $\pm$ 18.670
50	$\text{Linex}_{U}$	$0.348\pm 0.062$	237.603 $\pm$ 101.801	9.624 $\pm$ 18.677	9.730 $\pm$ 7.674	4.863 $\pm$ 4.338	27.847 $\pm$ 18.382
100	$\text{Linex}_{L}$	$0.193\pm 0.054$	1493.405 $\pm$ 345.786	0.264 $\pm$ 0.213	0.728 $\pm$ 1.420	2.834 $\pm$ 2.297	27.860 $\pm$ 18.211
100	$\text{Linex}_{U}$	$0.199\pm 0.060$	247.058 $\pm$ 57.860	0.268 $\pm$ 0.222	0.842 $\pm$ 1.519	2.834 $\pm$ 2.298	27.842 $\pm$ 17.782
250	$\text{Linex}_{L}$	$0.077\pm 0.017$	0.681 $\pm$ 1.673	0.034 $\pm$ 0.010	0.216 $\pm$ 0.295	1.127 $\pm$ 1.044	17.795 $\pm$ 11.973
250	$\text{Linex}_{U}$	$0.078\pm 0.017$	0.657 $\pm$ 1.782	0.032 $\pm$ 0.007	0.444 $\pm$ 0.542	1.121 $\pm$ 1.044	17.796 $\pm$ 11.975

Table 4: Ideal 1-dimensional PDE linear-exponential (Linex) error comparison across methods and training set sizes.

$n_{\text{train}}$	Metric	Naive	IBP	CROWN	INN	Mid-CROWN	Mid-IBP
50	PINAW	$1.247\pm 1.004$	$0.801\pm 0.299$	$1.516\pm 0.810$	$5.947\pm 8.095$	$4.452\pm 3.606$	$44.056\pm 6.963$
	PICP	$0.583\pm 0.224$	$0.151\pm 0.203$	$0.568\pm 0.274$	$0.724\pm 0.260$	$0.908\pm 0.171$	$0.996\pm 0.042$
	CWC	$13.097\pm 18.295$	$69.769\pm 44.149$	$25.230\pm 29.304$	$61.983\pm 236.542$	$11.717\pm 10.837$	$90.025\pm 27.421$
100	PINAW	$1.135\pm 0.502$	$0.782\pm 0.306$	$1.069\pm 0.123$	$1.155\pm 0.525$	$2.383\pm 1.224$	$55.671\pm 7.481$
	PICP	$0.730\pm 0.163$	$0.148\pm 0.201$	$0.674\pm 0.200$	$0.813\pm 0.156$	$0.912\pm 0.144$	$1.000\pm 0.007$
	CWC	$7.081\pm 8.472$	$68.333\pm 43.949$	$10.313\pm 12.772$	$6.474\pm 19.586$	$6.406\pm 4.013$	$111.421\pm 14.785$

Table 5: Ideal 1-dimensional PDE performance comparison across methods and training set sizes on three different metrics: predicted interval normalized average width (PINAW), predictied interval coverage probability (PICP), and the coverage width-based criterion (CWC).

$n_{\text{train}}$	Process	Naive	IBP	CROWN	INN	Mid-CROWN	Mid-IBP
50	Training	$127.201\pm 2.375$	$27.907\pm 0.573$	$236.503\pm 1.294$	$257.835\pm 5.166$	$235.662\pm 0.981$	$27.865\pm 0.446$
50	Inference	$0.972\pm 0.125$	$0.051\pm 0.002$	$0.502\pm 0.006$	$0.458\pm 0.030$	$0.506\pm 0.010$	$0.052\pm 0.001$
100	Training	$174.037\pm 8.729$	$28.411\pm 0.194$	$251.899\pm 1.708$	$442.901\pm 1.381$	$252.166\pm 2.666$	$28.886\pm 0.761$
100	Inference	$0.974\pm 0.103$	$0.050\pm 0.001$	$0.490\pm 0.005$	$0.457\pm 0.008$	$0.499\pm 0.005$	$0.072\pm 0.059$
250	Training	$276.729\pm 10.871$	$47.077\pm 0.177$	$508.158\pm 2.961$	$755.305\pm 18.820$	$517.749\pm 7.738$	$47.476\pm 0.804$
250	Inference	$1.059\pm 0.144$	$0.053\pm 0.002$	$0.527\pm 0.068$	$0.438\pm 0.010$	$0.499\pm 0.011$	$0.052\pm 0.001$

Table 6: Ideal 1-dimensional PDE time comparison in seconds (s). The IBP and CROWN methods are implemented using AutoLiRPA library in PyTorch framework. Naive and INN methods are impelented in TensorFlow framework due to the availability of existing codebases. The Opt-Prop method were not included in the time comparison due to its infeasibly high computational cost. All experiments were conducted on MacBook Pro with Apple M1 Pro chip.

Figure 6: A single input-output pair of an ideal interval dataset for a one-dimensional PDE.

$n_{\text{train}}$	Metric	Naive	IBP	CROWN	INN	Mid-CROWN	Mid-IBP
150	$\text{Linex}_{L}$	$0.148\pm 0.948$	$0.070\pm 0.248$	$0.056\pm 0.121$	$1.447\pm 22.186$	$0.127\pm 0.165$	$0.165\pm 0.213$
150	$\text{Linex}_{U}$	$0.141\pm 1.184$	$0.068\pm 0.238$	$0.054\pm 0.128$	$1.387\pm 28.283$	$0.126\pm 0.178$	$0.167\pm 0.217$
250	$\text{Linex}_{L}$	$0.062\pm 0.630$	$0.059\pm 0.162$	$0.044\pm 0.105$	$0.924\pm 11.532$	$0.114\pm 0.144$	$0.124\pm 0.158$
250	$\text{Linex}_{U}$	$0.061\pm 0.687$	$0.058\pm 0.157$	$0.045\pm 0.104$	$0.786\pm 6.351$	$0.113\pm 0.150$	$0.121\pm 0.177$
500	$\text{Linex}_{L}$	$0.027\pm 0.398$	$0.043\pm 0.102$	$0.034\pm 0.057$	$0.099\pm 0.489$	$0.088\pm 0.113$	$0.079\pm 0.098$
500	$\text{Linex}_{U}$	$0.027\pm 0.288$	$0.042\pm 0.092$	$0.034\pm 0.057$	$0.095\pm 0.527$	$0.085\pm 0.110$	$0.080\pm 0.099$

Table 7: Ideal 2-dimensional PDE linear-exponential (Linex) error comparison across methods and training set sizes.

$n_{\text{train}}$	Metric	Naive	IBP	CROWN	INN	Mid-CROWN	Mid-IBP
150	PINAW	$1.123\pm 0.272$	$1.706\pm 0.086$	$1.652\pm 0.137$	$2.908\pm 3.238$	$2.126\pm 0.216$	$2.459\pm 0.121$
	PICP	$0.518\pm 0.151$	$0.857\pm 0.130$	$0.842\pm 0.159$	$0.357\pm 0.310$	$0.949\pm 0.070$	$0.972\pm 0.033$
	CWC	$17.398\pm 14.332$	$6.389\pm 5.329$	$7.212\pm 8.503$	$114.530\pm 176.641$	$5.073\pm 1.778$	$5.326\pm 0.591$
250	PINAW	$1.060\pm 0.172$	$1.679\pm 0.081$	$1.574\pm 0.120$	$1.833\pm 1.617$	$2.087\pm 0.196$	$2.221\pm 0.109$
	PICP	$0.655\pm 0.133$	$0.873\pm 0.121$	$0.903\pm 0.109$	$0.413\pm 0.292$	$0.972\pm 0.037$	$0.970\pm 0.043$
	CWC	$8.684\pm 7.436$	$5.772\pm 4.255$	$4.710\pm 3.159$	$63.371\pm 102.673$	$4.523\pm 0.576$	$4.874\pm 0.866$

Table 8: Ideal 2-dimensional PDE performance comparison across methods and training set sizes on three different metrics: predicted interval normalized average width (PINAW), predictied interval coverage probability (PICP), and the coverage width-based criterion (CWC).

$n_{\text{train}}$	Process	Naive	IBP	CROWN	INN	Mid-CROWN	Mid-IBP
150	Training	$427.811\pm 5.657$	$157.825\pm 1.704$	$955.687\pm 21.946$	$194.951\pm 5.366$	$192.697\pm 3.074$	$159.386\pm 1.152$
150	Inference	$59.407\pm 2.461$	$3.972\pm 0.033$	$4.966\pm 0.167$	$62.753\pm 2.568$	$5.086\pm 0.090$	$3.938\pm 0.027$
250	Training	$654.959\pm 8.532$	$226.241\pm 4.122$	$275.351\pm 8.590$	$328.865\pm 3.390$	$277.282\pm 5.960$	$234.361\pm 5.051$
250	Inference	$49.773\pm 0.890$	$3.944\pm 0.026$	$5.216\pm 0.383$	$55.475\pm 2.059$	$5.326\pm 0.432$	$4.025\pm 0.085$
500	Training	$1048.473\pm 44.902$	$401.825\pm 9.772$	$469.127\pm 18.255$	$665.211\pm 14.863$	$462.445\pm 13.094$	$410.800\pm 8.188$
500	Inference	$53.315\pm 9.790$	$3.996\pm 0.047$	$4.859\pm 0.162$	$62.299\pm 8.123$	$4.868\pm 0.137$	$3.930\pm 0.026$

Table 9: Ideal 2-dimensional PDE time comparison in seconds (s). The IBP and CROWN methods are implemented using AutoLiRPA library in PyTorch framework. Naive and INN methods are impelented in TensorFlow framework due to the availability of existing codebases. The Opt-Prop method were not included in the time comparison due to its infeasibly high computational cost. All experiments were conducted on NVidia P100 GPU.

Panel of Figure 7 — Figure 7: Illustration on (a) Pointwise training dataset with interval test data. (b) Augmented interval training data generated from grid-based interval augmentation.

$n_{\text{train}}$	Metric	Naive	IBP	CROWN	INN	Mid-CROWN	Mid-IBP	Opt-Prop
10	$\text{RMSE}_{L}$	0.197 $\pm$ 0.140	0.147 $\pm$ 0.075	0.118 $\pm$ 0.053	0.167 $\pm$ 0.075	0.125 $\pm$ 0.095	0.458 $\pm$ 0.166	0.118 $\pm$ 0.170
10	$\text{RMSE}_{U}$	0.321 $\pm$ 0.195	0.170 $\pm$ 0.077	0.137 $\pm$ 0.060	0.125 $\pm$ 0.055	0.149 $\pm$ 0.095	0.362 $\pm$ 0.122	0.115 $\pm$ 0.141
25	$\text{RMSE}_{L}$	0.065 $\pm$ 0.018	0.087 $\pm$ 0.017	0.064 $\pm$ 0.011	0.062 $\pm$ 0.014	0.059 $\pm$ 0.016	0.356 $\pm$ 0.115	0.052 $\pm$ 0.007
25	$\text{RMSE}_{U}$	0.082 $\pm$ 0.038	0.117 $\pm$ 0.025	0.087 $\pm$ 0.035	0.067 $\pm$ 0.025	0.067 $\pm$ 0.033	0.339 $\pm$ 0.115	0.047 $\pm$ 0.006
50	$\text{RMSE}_{L}$	0.043 $\pm$ 0.009	0.091 $\pm$ 0.016	0.055 $\pm$ 0.009	0.055 $\pm$ 0.009	0.047 $\pm$ 0.009	0.326 $\pm$ 0.058	0.045 $\pm$ 0.007
50	$\text{RMSE}_{U}$	0.058 $\pm$ 0.015	0.119 $\pm$ 0.019	0.060 $\pm$ 0.010	0.075 $\pm$ 0.038	0.044 $\pm$ 0.011	0.328 $\pm$ 0.063	0.050 $\pm$ 0.006

Table 10: Augmented 1-dimensional regression RMSE comparison across methods and training set sizes.

$n_{\text{train}}$	Metric	Naive	IBP	CROWN	INN	Mid-CROWN	Mid-IBP	Opt-Prop
10	PINAW	$0.144\pm 0.940$	$0.539\pm 0.071$	$0.467\pm 0.037$	$0.511\pm 0.624$	$0.535\pm 0.061$	$4.598\pm 1.417$	$0.478\pm 0.102$
	PICP	$0.399\pm 0.086$	$0.385\pm 0.077$	$0.396\pm 0.031$	$0.437\pm 0.134$	$0.418\pm 0.101$	$0.937\pm 0.042$	$0.399\pm 0.053$
	CWC	$3.241\pm 23.543$	$12.694\pm 3.873$	$10.088\pm 1.088$	$5.557\pm 17.497$	$11.573\pm 5.795$	$10.820\pm 2.684$	$10.827\pm 5.546$
25	PINAW	$0.604\pm 0.223$	$0.498\pm 0.032$	$0.421\pm 0.025$	$0.635\pm 0.138$	$0.523\pm 0.041$	$4.450\pm 1.332$	$0.462\pm 0.018$
	PICP	$0.527\pm 0.081$	$0.348\pm 0.062$	$0.400\pm 0.030$	$0.540\pm 0.077$	$0.481\pm 0.064$	$0.988\pm 0.014$	$0.453\pm 0.016$
	CWC	$6.665\pm 1.226$	$14.043\pm 3.788$	$8.878\pm 0.776$	$6.973\pm 1.361$	$7.785\pm 2.276$	$9.138\pm 2.606$	$7.573\pm 0.367$

Table 11: Augmented 1-dimensional regression performance comparison across methods and training set sizes on three different metrics: predicted interval normalized average width (PINAW), predictied interval coverage probability (PICP), and the coverage width-based criterion (CWC).

Panel of Figure 8 — Figure 8: Augmentation result example of a single cluster. (a) Augmented input function. (b) Augmented output function.

Figure 9: Augmented input function (left) and comparison between the augmented output and the ground truth output propagated through PDE solver.

$n_{\text{train}}$	Metric	Naive	IBP ∗	CROWN	INN ∗	Mid-CROWN	Mid-IBP
50	$\text{Linex}_{L}$	$1.559\pm 0.495$	2137.172 $\pm$ 1059.331	20.695 $\pm$ 48.603	89.456 $\pm$ 92.321	0.341 $\pm$ 0.083	82.23 $\pm$ 85.715
50	$\text{Linex}_{U}$	$1.610\pm 0.653$	898.399 $\pm$ 442.959	40.634 $\pm$ 106.411	260.384 $\pm$ 299.820	0.390 $\pm$ 0.090	42.949 $\pm$ 39.487
100	$\text{Linex}_{L}$	$1.191\pm 0.210$	2082.647 $\pm$ 914.356	3.815 $\pm$ 1.002	22.755 $\pm$ 15.753	0.282 $\pm$ 0.075	82.573 $\pm$ 98.971
100	$\text{Linex}_{U}$	$1.354\pm 0.424$	856.115 $\pm$ 345.489	3.285 $\pm$ 1.435	34.114 $\pm$ 31.953	0.469 $\pm$ 0.554	25.470 $\pm$ 22.488
250	$\text{Linex}_{L}$	$0.925\pm 0.149$	2139.45 $\pm$ 560.089	3.496 $\pm$ 0.764	5.860 $\pm$ 2.474	0.162 $\pm$ 0.043	29.698 $\pm$ 22.862
250	$\text{Linex}_{U}$	$1.125\pm 0.198$	745.528 $\pm$ 184.611	3.486 $\pm$ 0.685	8.732 $\pm$ 3.769	0.168 $\pm$ 0.067	10.974 $\pm$ 2.447

Table 12: Linear-exponential (Linex) error of augmented 1D PDE experiments across methods and training sizes. For INN and IBP, the asterisk ( ∗ ) denotes cases where some runs failed to converge or produced NaN values; results are averaged over successful runs only.

$n_{\text{train}}$	Metric	Naive	IBP ∗	CROWN	INN ∗	Mid-CROWN	Mid-IBP
50	PINAW	$1.231\pm 0.947$	$0.362\pm 0.034$	$0.762\pm 0.024$	$1121.507\pm 3416.796$	$1.380\pm 0.266$	$3.647\pm 1.540$
	PICP	$0.390\pm 0.193$	$0.069\pm 0.089$	$0.086\pm 0.076$	$1.000\pm 0.000$	$0.708\pm 0.279$	$0.649\pm 0.407$
	CWC	$37.297\pm 46.336$	$41.290\pm 13.860$	$78.941\pm 24.501$	$2243.015\pm 6833.591$	$18.418\pm 32.571$	$67.272\pm 89.325$
100	PINAW	$1.081\pm 0.924$	$0.365\pm 0.018$	$0.702\pm 0.025$	$110.540\pm 1580.054$	$1.378\pm 0.156$	$3.646\pm 1.484$
	PICP	$0.378\pm 0.234$	$0.069\pm 0.087$	$0.257\pm 0.179$	$0.707\pm 0.292$	$0.808\pm 0.227$	$0.668\pm 0.410$
	CWC	$39.501\pm 67.005$	$41.766\pm 13.637$	$41.301\pm 30.647$	$318.042\pm 3165.910$	$10.119\pm 19.879$	$65.564\pm 93.063$

Table 13: Augmented 1-dimensional PDE performance comparison across methods and training set sizes on three different metrics: predicted interval normalized average width (PINAW), predictied interval coverage probability (PICP), and the coverage width-based criterion (CWC). For INN and IBP, the asterisk ( ∗ ) denotes cases where some runs failed to converge or produced NaN values; results are averaged over successful runs only.

$n_{\text{train}}$	Metric	Naive	IBP	CROWN	INN	Mid-CROWN	Mid-IBP
150	$\text{Linex}_{L}$	0.224 $\pm$ 1.946	0.207 $\pm$ 0.590	0.140 $\pm$ 0.857	12.876 $\pm$ 1.171	0.086 $\pm$ 0.018	0.075 $\pm$ 0.021
150	$\text{Linex}_{U}$	0.212 $\pm$ 1.076	0.192 $\pm$ 0.526	0.131 $\pm$ 0.616	9.832 $\pm$ 0.868	0.089 $\pm$ 0.032	0.081 $\pm$ 0.011
250	$\text{Linex}_{L}$	0.119 $\pm$ 0.063	0.175 $\pm$ 0.049	0.102 $\pm$ 0.016	14.413 $\pm$ 1.214	0.074 $\pm$ 0.003	0.060 $\pm$ 0.005
250	$\text{Linex}_{U}$	0.111 $\pm$ 0.078	0.166 $\pm$ 0.048	0.086 $\pm$ 0.010	9.022 $\pm$ 1.201	0.078 $\pm$ 0.008	0.059 $\pm$ 0.005
500	$\text{Linex}_{L}$	0.066 $\pm$ 0.003	0.103 $\pm$ 0.021	0.067 $\pm$ 0.007	1.728 $\pm$ 0.651	0.065 $\pm$ 0.007	0.042 $\pm$ 0.003
500	$\text{Linex}_{U}$	0.064 $\pm$ 0.003	0.100 $\pm$ 0.022	0.060 $\pm$ 0.006	1.317 $\pm$ 0.661	0.069 $\pm$ 0.005	0.044 $\pm$ 0.003

Table 14: Augmented 2-dimensional PDE linear-exponential (Linex) error comparison across methods and training set sizes.

$n_{\text{train}}$	Metric	Naive	IBP	CROWN	INN	Mid-CROWN	Mid-IBP
150	PINAW	$1.671\pm 0.563$	$1.488\pm 0.072$	$1.536\pm 0.130$	$3.604\pm 3.465$	$1.911\pm 0.194$	$1.793\pm 0.090$
	PICP	$0.612\pm 0.177$	$0.734\pm 0.161$	$0.737\pm 0.206$	$0.285\pm 0.192$	$0.911\pm 0.105$	$0.909\pm 0.091$
	CWC	$20.748\pm 26.414$	$9.886\pm 10.313$	$11.946\pm 14.837$	$134.387\pm 116.458$	$5.429\pm 2.913$	$5.024\pm 2.248$
250	PINAW	$1.558\pm 0.430$	$1.519\pm 0.073$	$1.511\pm 0.095$	$3.157\pm 2.823$	$1.879\pm 0.199$	$1.733\pm 0.083$
	PICP	$0.710\pm 0.157$	$0.636\pm 0.197$	$0.818\pm 0.159$	$0.266\pm 0.180$	$0.907\pm 0.110$	$0.911\pm 0.105$
	CWC	$11.877\pm 14.653$	$17.284\pm 19.516$	$7.037\pm 7.087$	$133.544\pm 108.901$	$5.510\pm 3.513$	$5.116\pm 4.350$

Table 15: Augmented 2-dimensional PDE performance comparison across methods and training set sizes on three different metrics: predicted interval normalized average width (PINAW), predictied interval coverage probability (PICP), and the coverage width-based criterion (CWC).

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Configuration	Metric	Ideal		Augmented
Configuration	Metric	Naive	INN	Naive	Mid-CROWN
2L-8	$\text{RMSE}_{L}$	$0.050\pm 0.057$	$0.043\pm 0.041$	$0.065\pm 0.023$	$0.087\pm 0.028$
2L-8	$\text{RMSE}_{U}$	$0.058\pm 0.062$	$0.040\pm 0.044$	$0.067\pm 0.032$	$0.104\pm 0.043$
2L-16	$\text{RMSE}_{L}$	$0.034\pm 0.041$	$0.043\pm 0.039$	$0.062\pm 0.036$	$0.069\pm 0.017$
2L-16	$\text{RMSE}_{U}$	$0.047\pm 0.053$	$0.034\pm 0.035$	$0.075\pm 0.040$	$0.077\pm 0.030$
2L-32	$\text{RMSE}_{L}$	$0.021\pm 0.006$	$0.031\pm 0.019$	$0.046\pm 0.005$	$0.063\pm 0.007$

Appendix

Table 16: Brief hyperparameter study of neural network for the 1D regression, focusing on the two best-performing methods in each experimental setting on

n_{\text{train}}=50

. The configuration “2L-8” notation denotes that both the branch and trunk networks consist of 2 hidden layers with 8 neurons per layer.

Configuration	Metric	Ideal		Augmented
Configuration	Metric	Naive	CROWN	Naive	Mid-CROWN
2L-32	$\text{Linex}_{L}$	$0.108\pm 0.045$	$0.019\pm 0.005$	$0.912\pm 0.056$	$1.749\pm 0.235$
2L-32	$\text{Linex}_{U}$	$0.107\pm 0.045$	$0.019\pm 0.004$	$0.798\pm 0.069$	$1.704\pm 0.243$
2L-64	$\text{Linex}_{L}$	$0.152\pm 0.072$	$0.012\pm 0.005$	$0.904\pm 0.070$	$1.223\pm 0.194$
2L-64	$\text{Linex}_{U}$	$0.154\pm 0.073$	$0.011\pm 0.005$	$0.839\pm 0.095$	$1.179\pm 0.228$
2L-128	$\text{Linex}_{L}$	$0.173\pm 0.082$	$0.009\pm 0.002$	$0.946\pm 0.098$	$0.713\pm 0.099$

Appendix

Table 17: Brief hyperparameter study of DeepONet for the 1D PDE, focusing on the two best-performing methods in each experimental setting on

n_{\text{train}}=500

. The configuration “2L-32” notation denotes that both the branch and trunk networks consist of 2 hidden layers with 32 neurons per layer.

Uncertainty Propagation

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A numerical study into neural network surrogate model performance for uncertainty propagation

Noah Wade, Kirubel Teferra

ASEE Postdoctoral Associate, U.S. Naval Research Laboratory, Washington, DC, 20375, USA · U.S. Naval Research Laboratory, Washington, 20375, DC, USA

Neural Surrogate Modeling Uncertainty Quantification PDE Surrogate Modeling OOD Generalization Deep Operator Networks Uncertainty Propagation

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Neural network surrogate models have emerged as a promising approach to model solution fields for a wide variety of boundary value problems encountered in physical modeling. Stochastic problems represent an area of particularly high interest because of the potential to significantly reduce the repeated evaluation of expensive forward models via traditional numerical solvers when conducting parametric analysis. However, many studies found in the literature primarily focus on the ability of neural network surrogate models to represent deterministic samples or mean field solutions and largely overlook surrogate model performance at the tails of the distribution. The present study examines in detail the ability of neural network surrogate models to capture the full distribution of solution fields over the entire probability space, while emphasis is placed at the tails of the distribution. Serving as a canonical problem is the heat conduction equation with a highly stochastic source term, inducing extremely large variation in the thermal solution field. Comparisons are made between a classic feed-forward fully connected network and a Deep Operator Network architecture, using both data-driven and physics-informed loss functions. Results show that the worst-case prediction errors are an order of magnitude larger than the mean field error, highlighting the importance of the outlier samples. The large errors associated with extreme samples result from the networks having to extrapolate beyond the bounds of the training data. A method for identifying these samples is presented along with a discussion of potential approaches to account of their errors. Among the models considered, the fully connected neural network trained using a weak form residual loss performs best in handling these extrapolated inputs, achieving the highest prediction accuracy for the numerically produced datasets.

Abstract

Uncertainty Propagation

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