Distribution-Free Robust Functional Predict-Then-Optimize
This addresses uncertainty quantification for neural operators in PDE-based decision-making, offering a scalable, distribution-free solution with practical applications in engineering.
The paper tackles the problem that neural operator surrogate models for PDEs lack calibrated uncertainty quantification, proposing a distribution-free conformal prediction method that provides uncertainty regions over function spaces and enables formal regret characterization in downstream robust decision-making tasks, with empirical demonstrations showing superior performance over restrictive methods like Gaussian Processes.
The solution of PDEs in decision-making tasks is increasingly being undertaken with the help of neural operator surrogate models due to the need for repeated evaluation. Such methods, while significantly more computationally favorable compared to their numerical counterparts, fail to provide any calibrated notions of uncertainty in their predictions. Current methods approach this deficiency typically with ensembling or Bayesian posterior estimation. However, these approaches either require distributional assumptions that fail to hold in practice or lack practical scalability, limiting their applications in practice. We, therefore, propose a novel application of conformal prediction to produce distribution-free uncertainty quantification over the function spaces mapped by neural operators. We then demonstrate how such prediction regions enable a formal regret characterization if leveraged in downstream robust decision-making tasks. We further demonstrate how such posited robust decision-making tasks can be efficiently solved using an infinite-dimensional generalization of Danskin's Theorem and calculus of variations and empirically demonstrate the superior performance of our proposed method over more restrictive modeling paradigms, such as Gaussian Processes, across several engineering tasks.