Deep set based operator learning with uncertainty quantification
This addresses limitations in operator learning for scientific machine learning by enabling uncertainty quantification and flexible sensor handling, though it builds incrementally on existing permutation-invariant methods.
The paper tackled the problem of learning operators from data with uncertainty quantification, proposing UQ-SONet to handle sparse and variable sensor locations while providing principled uncertainty estimation, achieving robustness and effectiveness in numerical experiments on deterministic and stochastic PDEs like the Navier-Stokes equation.
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.