Deep set based operator learning with uncertainty quantification

• URL: http://arxiv.org/abs/2509.25646v1
• Date: Tue, 30 Sep 2025 01:36:18 GMT
• Title: Deep set based operator learning with uncertainty quantification
• Abstract summary: We propose UQ-SONet, a permutation-invariant operator learning framework with built-in uncertainty quantification (UQ)<n>Our model integrates a set transformer embedding to handle sparse and variable sensor locations, and employs a conditional variational autoencoder (cVAE) to approximate the conditional distribution of the solution operator.
• Score: 14.436821529642893
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Learning operators from data is central to scientific machine learning. While DeepONets are widely used for their ability to handle complex domains, they require fixed sensor numbers and locations, lack mechanisms for uncertainty quantification (UQ), and are thus limited in practical applicability. Recent permutationinvariant extensions, such as the Variable-Input Deep Operator Network (VIDON), relax these sensor constraints but still rely on sufficiently dense observations and cannot capture uncertainties arising from incomplete measurements or from operators with inherent randomness. To address these challenges, we propose UQ-SONet, a permutation-invariant operator learning framework with built-in UQ. Our model integrates a set transformer embedding to handle sparse and variable sensor locations, and employs a conditional variational autoencoder (cVAE) to approximate the conditional distribution of the solution operator. By minimizing the negative ELBO, UQ-SONet provides principled uncertainty estimation while maintaining predictive accuracy. Numerical experiments on deterministic and stochastic PDEs, including the Navier-Stokes equation, demonstrate the robustness and effectiveness of the proposed framework.

Related papers

• Diffusion-Based Stochastic Operator Networks for Uncertainty Quantification in Stochastic Partial Differential Equations [0.3290514527479967]
  We introduce a novel framework for uncertainty quantification of solution operators associated with partial differential equations (SPDEs)<n>The proposed method, namely the Operator Network (SON), is constructed by combining the structure of the Deep Operator Network (DeepONet) with Neural Networks (SNNs)<n> Numerical experiments on benchmark SPDEs under multiple uncertainty sources demonstrate the accuracy and robustness of the proposed method.
  arXiv  Detail & Related papers  (2026-05-16T18:19:45Z)
• Structure-Aware Epistemic Uncertainty Quantification for Neural Operator PDE Surrogates [10.39662017167378]
  Uncertainty quantification (UQ) must be both computationally efficient and spatially faithful.<n>We propose a structure-aware UQ scheme that exploits the modular anatomy common to modern NOs.<n> Experiments on challenging PDE benchmarks demonstrate that the proposed structure-aware design yields more reliable coverage, tighter bands, and improved residual-uncertainty alignment.
  arXiv  Detail & Related papers  (2026-02-24T10:17:24Z)
• Equivariant Evidential Deep Learning for Interatomic Potentials [55.6997213490859]
  Uncertainty quantification is critical for assessing the reliability of machine learning interatomic potentials in molecular dynamics simulations.<n>Existing UQ approaches for MLIPs are often limited by high computational cost or suboptimal performance.<n>We propose textitEquivariant Evidential Deep Learning for Interatomic Potentials ($texte2$IP), a backbone-agnostic framework that models atomic forces and their uncertainty jointly.
  arXiv  Detail & Related papers  (2026-02-11T02:00:25Z)
• Multidimensional Uncertainty Quantification via Optimal Transport [87.97146725546502]
  We take a multidimensional view on uncertainty quantification (UQ) by stacking complementary UQ measures into a vector.<n>VecUQ-OT shows high efficiency even when individual measures fail.
  arXiv  Detail & Related papers  (2025-09-26T14:09:03Z)
• Light-Weight Diffusion Multiplier and Uncertainty Quantification for Fourier Neural Operators [1.315766151337659]
  We introduce DINOZAUR: a diffusion-based neural operator parametrization with uncertainty quantification.<n>Our method achieves competitive or superior performance across several PDE benchmarks.
  arXiv  Detail & Related papers  (2025-08-01T13:57:19Z)
• COIN: Uncertainty-Guarding Selective Question Answering for Foundation Models with Provable Risk Guarantees [51.5976496056012]
  COIN is an uncertainty-guarding selection framework that calibrates statistically valid thresholds to filter a single generated answer per question.<n>COIN estimates the empirical error rate on a calibration set and applies confidence interval methods to establish a high-probability upper bound on the true error rate.<n>We demonstrate COIN's robustness in risk control, strong test-time power in retaining admissible answers, and predictive efficiency under limited calibration data.
  arXiv  Detail & Related papers  (2025-06-25T07:04:49Z)
• SetONet: A Deep Set-based Operator Network for Solving PDEs with permutation invariant variable input sampling [2.95983424663256]
  We introduce the Set Operator Network (SetONet), a novel architecture that integrates Deep Sets principles into the DeepONet framework.<n>The core innovation lies in the SetONet branch network, which processes the input function as an unordered emphset of location-value pairs.<n>We demonstrate SetONet's effectiveness on several benchmark problems, including derivative/anti-derivative operators, 1D Darcy flow, and 2D elasticity.
  arXiv  Detail & Related papers  (2025-05-07T18:50:05Z)
• Distribution free uncertainty quantification in neuroscience-inspired deep operators [1.8416014644193066]
  Energy-efficient deep learning algorithms are essential for a sustainable future and feasible edge computing setups.<n>In this paper, we introduce the Conformalized Randomized Prior Operator (CRP-O) framework to quantify uncertainty in both conventional and spiking neural operators.<n>We show that the conformalized RP-VSWNO significantly enhance UQ estimates compared to vanilla RP-VSWNO, Quantile WNO (Q-WNO), and Conformalized Quantile WNO (CQ-WNO)
  arXiv  Detail & Related papers  (2024-12-12T15:37:02Z)
• Accelerated zero-order SGD under high-order smoothness and overparameterized regime [79.85163929026146]
  We present a novel gradient-free algorithm to solve convex optimization problems. Such problems are encountered in medicine, physics, and machine learning. We provide convergence guarantees for the proposed algorithm under both types of noise.
  arXiv  Detail & Related papers  (2024-11-21T10:26:17Z)
• Alpha-VI DeepONet: A prior-robust variational Bayesian approach for enhancing DeepONets with uncertainty quantification [0.0]
  We introduce a novel deep operator network (DeepONet) framework that incorporates generalised variational inference (GVI) By incorporating Bayesian neural networks as the building blocks for the branch and trunk networks, our framework endows DeepONet with uncertainty quantification. We demonstrate that modifying the variational objective function yields superior results in terms of minimising the mean squared error.
  arXiv  Detail & Related papers  (2024-08-01T16:22:03Z)
• Uncertainty Quantification for Forward and Inverse Problems of PDEs via Latent Global Evolution [110.99891169486366]
  We propose a method that integrates efficient and precise uncertainty quantification into a deep learning-based surrogate model. Our method endows deep learning-based surrogate models with robust and efficient uncertainty quantification capabilities for both forward and inverse problems. Our method excels at propagating uncertainty over extended auto-regressive rollouts, making it suitable for scenarios involving long-term predictions.
  arXiv  Detail & Related papers  (2024-02-13T11:22:59Z)
• IB-UQ: Information bottleneck based uncertainty quantification for neural function regression and neural operator learning [11.5992081385106]
  We propose a novel framework for uncertainty quantification via information bottleneck (IB-UQ) for scientific machine learning tasks. We incorporate the bottleneck by a confidence-aware encoder, which encodes inputs into latent representations according to the confidence of the input data. We also propose a data augmentation based information bottleneck objective which can enhance the quality of the extrapolation uncertainty.
  arXiv  Detail & Related papers  (2023-02-07T05:56:42Z)
• Robust Implicit Networks via Non-Euclidean Contractions [63.91638306025768]
  Implicit neural networks show improved accuracy and significant reduction in memory consumption. They can suffer from ill-posedness and convergence instability. This paper provides a new framework to design well-posed and robust implicit neural networks.
  arXiv  Detail & Related papers  (2021-06-06T18:05:02Z)
• Scalable Uncertainty for Computer Vision with Functional Variational Inference [18.492485304537134]
  We leverage the formulation of variational inference in function space. We obtain predictive uncertainty estimates at the cost of a single forward pass through any chosen CNN architecture. We propose numerically efficient algorithms which enable fast training in the context of high-dimensional tasks.
  arXiv  Detail & Related papers  (2020-03-06T19:09:42Z)
• Uncertainty Estimation Using a Single Deep Deterministic Neural Network [66.26231423824089]
  We propose a method for training a deterministic deep model that can find and reject out of distribution data points at test time with a single forward pass. We scale training in these with a novel loss function and centroid updating scheme and match the accuracy of softmax models.
  arXiv  Detail & Related papers  (2020-03-04T12:27:36Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.