Operator learning is a powerful paradigm for solving partial differential equations, with Fourier Neural Operators serving as a widely adopted foundation. However, FNOs face significant scalability challenges due to overparameterization and offer no native uncertainty quantification -- a key requirement for reliable scientific and engineering applications. Instead, neural operators rely on post hoc UQ methods that ignore geometric inductive biases. In this work, we introduce DINOZAUR: a diffusion-based neural operator parametrization with uncertainty quantification. Inspired by the structure of the heat kernel, DINOZAUR replaces the dense tensor multiplier in FNOs with a dimensionality-independent diffusion multiplier that has a single learnable time parameter per channel, drastically reducing parameter count and memory footprint without compromising predictive performance. By defining priors over those time parameters, we cast DINOZAUR as a Bayesian neural operator to yield spatially correlated outputs and calibrated uncertainty estimates. Our method achieves competitive or superior performance across several PDE benchmarks while providing efficient uncertainty quantification.

Intelligent buildings are equipped with sensing systems able to measure the contaminant concentration in the different building zones for safety purposes. The aim of these systems is to promptly detect the presence of a contaminant so that appropriate actions can be taken to ensure the safety of the people. At the same time, these sensing systems, which operate in real-world conditions, suffer from noise and sensor degradation faults. Both noise and faults can induce false alarms (resulting in unnecessary disruptive actions such as building evacuation) or missed alarms (when the presence of a contaminant is not detected). This paper proposes a novel cognitive monitoring system for performing contaminant detection in intelligent buildings with real-time point-trigger sensors. The proposed system reduces the occurrence of false alarms by means of a three-layered architecture, which employs cognitive mechanisms to validate possible detections and discriminate between the presence of a real contaminant source and a degradation fault affecting the sensors of the sensing system. In addition, the proposed system is able to isolate the building zone containing the contaminant source (or the faulty sensor) and estimate the onset time of the release (or the fault).