We propose a method for obtaining statistically guaranteed prediction sets for functional machine learning methods: surrogate models which map between function spaces, motivated by the need to build reliable PDE emulators. The method constructs nested prediction sets on a low-dimensional representation (an SVD) of the surrogate model’s error, and then maps these sets to the prediction space using set-propagation techniques. This results in prediction sets for functional surrogate models with conformal prediction coverage guarantees. We use zonotopes as basis of the set construction, which allow an exact linear propagation and are closed under Cartesian products, making them well-suited to this high-dimensional problem. The method is model agnostic and can thus be applied to complex Sci-ML models, including Neural Operators, but also in simpler settings. We also introduce a technique to capture the truncation error of the SVD, preserving the guarantees of the method.

Neural PDEs have emerged as inexpensive surrogate models for numerical PDE solvers. While they offer efficient approximations, they often lack robust uncertainty quantification (UQ), limiting their practical utility. Existing UQ methods for these models typically have high computational demands and lack guarantees. We introduce a novel framework for calibrated physics-informed uncertainty quantification to address these limitations. Our approach leverages physics residual errors as a nonconformity score within a conformal prediction (CP) framework. This enables data-free, model-agnostic, and statistically guaranteed uncertainty estimates. Our framework utilises convolutional layers as finite difference stencils for gradient estimation, our framework provides inexpensive coverage bounds for the violation of conservation laws within model predictions. In our experiments, we utilise CP to obtain marginal coverage for each cell and joint coverage over the entire prediction domain of various PDEs.

This article focuses on predicting how an object can be transformed to a semantically meaningful pose relative to another object, given only one or few examples. Current pose correspondence methods rely on vast 3D object datasets and do not actively consider semantic information, which limits the objects to which they can be applied. We present a novel method for learning cross-object pose correspondence. The proposed method detects interacting object parts, performs one-shot part correspondence, and uses geometric and visual-semantic features. Given one example of two objects posed relative to each other, the model can learn how to transfer the demonstrated relations to unseen object instances.

Learning long-term behaviors in chaotic dynamical systems, such as turbulent flows and climate modelling, is challenging due to their inherent instability and unpredictability. These systems exhibit positive Lyapunov exponents, which significantly hinder accurate long-term forecasting. As a result, understanding long-term statistical behavior is far more valuable than focusing on short-term accuracy. While autoregressive deep sequence models have been applied to capture long-term behavior, they often lead to exponentially increasing errors in learned dynamics. To address this, we shift the focus from simple prediction errors to preserving an invariant measure in dissipative chaotic systems. These systems have attractors, where trajectories settle, and the invariant measure is the probability distribution on attractors that remains unchanged under dynamics. Existing methods generate long trajectories of dissipative chaotic systems by aligning invariant measures, but it is not always possible to obtain invariant measures for arbitrary datasets. We propose the Poincaré Flow Neural Network (PFNN), a novel operator learning framework designed to capture behaviors of chaotic systems without any explicit knowledge of the invariant measure. PFNN employs an auto-encoder to map the chaotic system to a finite-dimensional feature space, effectively linearizing the chaotic evolution. It then learns the linear evolution operators to match the physical dynamics by addressing two critical properties in dissipative chaotic systems (1) contraction, the system’s convergence toward its attractors, and (2) measure invariance, trajectories on the attractors following a probability distribution invariant to the dynamics. Our experiments on a variety of chaotic systems, including Lorenz systems, Kuramoto-Sivashinsky equation and Navier–Stokes equation, demonstrate that PFNN has more accurate predictions and physical statistics compared to competitive baselines including the Fourier Neural Operator and the Markov Neural Operator.