Deep set based operator learning with uncertainty quantification

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 permutation-invariant extensions, such as the V ariable-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. T o 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 (cV AE) 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. Introduction Learning continuous operators or complex systems from scattered data streams has shown promising success in scientific machine learning, which focuses on modeling mappings between infinite-dimensional function spaces. A prominent example is the Fourier Neural Operator (FNO) [1], which parameterizes the integral kernel in Fourier space. While highly effective for problems on domains that can be discretized or efficiently mapped to Cartesian grids, the applicability of FNO to more general settings remains limited [2].