Neural operators aim to approximate the solution operator of a system of differential equations purely from data. They have shown immense success in modeling complex dynamical systems across various domains. However, the occurrence of uncertainties inherent in both model and data has so far rarely been taken into account\textemdash{}a critical limitation in complex, chaotic systems such as weather forecasting. In this paper, we introduce the probabilistic neural operator (PNO), a framework for learning probability distributions over the output function space of neural operators. PNO extends neural operators with generative modeling based on strictly proper scoring rules, integrating uncertainty information directly into the training process. We provide a theoretical justification for the approach and demonstrate improved performance in quantifying uncertainty across different domains and with respect to different baselines. Furthermore, PNO requires minimal adjustment to existing architectures, shows improved performance for most probabilistic prediction tasks, and leads to well-calibrated predictive distributions and adequate uncertainty representations even for long dynamical trajectories. Implementing our approach into large-scale models for physical applications can lead to improvements in corresponding uncertainty quantification and extreme event identification, ultimately leading to a deeper understanding of the prediction of such surrogate models.

Recent advancements in machine learning have transformed the discovery of physical laws, moving from manual derivation to data-driven methods that simultaneously learn both the structure and parameters of governing equations. This shift introduces new challenges regarding the validity of the discovered equations, particularly concerning their uniqueness and, hence, identifiability. While the issue of non-uniqueness has been well-studied in the context of parameter estimation, it remains underexplored for algorithms that recover both structure and parameters simultaneously. Early studies have primarily focused on idealized scenarios with perfect, noise-free data. In contrast, this paper investigates how noise influences the uniqueness and identifiability of physical laws governed by partial differential equations (PDEs). We develop a comprehensive mathematical framework to analyze the uniqueness of PDEs in the presence of noise and introduce new algorithms that account for noise, providing thresholds to assess uniqueness and identifying situations where excessive noise hinders reliable conclusions. Numerical experiments demonstrate the effectiveness of these algorithms in detecting uniqueness despite the presence of noise.

We study the generalization capabilities of Message Passing Neural Networks (MPNNs), a prevalent class of Graph Neural Networks (GNN). We derive generalization bounds specifically for MPNNs with normalized sum aggregation and mean aggregation. Our analysis is based on a data generation model incorporating a finite set of template graphons. Each graph within this framework is generated by sampling from one of the graphons with a certain degree of perturbation. In particular, we extend previous MPNN generalization results to a more realistic setting, which includes the following modifications: 1) we analyze simple random graphs with Bernoulli-distributed edges instead of weighted graphs; 2) we sample both graphs and graph signals from perturbed graphons instead of clean graphons; and 3) we analyze sparse graphs instead of dense graphs. In this more realistic and challenging scenario, we provide a generalization bound that decreases as the average number of nodes in the graphs increases. Our results imply that MPNNs with higher complexity than the size of the training set can still generalize effectively, as long as the graphs are sufficiently large.

For example, in organic chemistry or bioinformatics, different types of cycles can impact We introduce r-loopy Weisfeiler-Leman (r-lWL), various chemical properties of the underlying molecules a novel hierarchy of graph isomorphism tests and (Deshpande et al., 2002; Koyutürk et al., 2004). Therefore, a corresponding GNN framework, r-lMPNN, that it is crucial to investigate whether GNNs can count certain can count cycles up to length r + 2. Most notably, substructures and to design architectures that surpass the we show that r-lWL can count homomorphisms limited power of MPNNs. of cactus graphs. This strictly extends classical Many models have been proposed to match or surpass the 1-WL, which can only count homomorphisms of expressive power of WL. Several draw inspiration from trees and, in fact, is incomparable to k-WL for any higher-order variants of the WL algorithm (Morris et al., fixed k. We empirically validate the expressive 2019), enabling them to count a broader range of substructures.

Solving these equations analytically is often challenging or even impossible, necessitating the utilization of other methods to obtain approximate solutions. One way to find approximate solutions to partial differential equations is through classical numerical methods. These methods have been studied for years and already have strong theoretical foundations when it comes to error estimation [1]. However, in recent years, with the rise of machine learning as a whole, there has also been an increased interest in applying machine learning methods to the problem of finding approximate solutions to PDEs. As universal function approximators [2], deep neural networks provide a promising avenue for a multitude of approaches to the approximation of solutions to partial differential equations. Among these methods are neural operators, methods based on the Feynman-Kac formula, and methods for parametric PDEs [3] [4] [5]. This paper focuses on physics-informed neural networks (PINNs), which were conceived as feed-forward neural networks that incorporate the dynamics of the PDE into their loss function [6].

Deep learning still has drawbacks in terms of trustworthiness, which describes a comprehensible, fair, safe, and reliable method. To mitigate the potential risk of AI, clear obligations associated to trustworthiness have been proposed via regulatory guidelines, e.g., in the European AI Act. Therefore, a central question is to what extent trustworthy deep learning can be realized. Establishing the described properties constituting trustworthiness requires that the factors influencing an algorithmic computation can be retraced, i.e., the algorithmic implementation is transparent. Motivated by the observation that the current evolution of deep learning models necessitates a change in computing technology, we derive a mathematical framework which enables us to analyze whether a transparent implementation in a computing model is feasible. We exemplarily apply our trustworthiness framework to analyze deep learning approaches for inverse problems in digital and analog computing models represented by Turing and Blum-Shub-Smale Machines, respectively. Based on previous results, we find that Blum-Shub-Smale Machines have the potential to establish trustworthy solvers for inverse problems under fairly general conditions, whereas Turing machines cannot guarantee trustworthiness to the same degree.

This work bridges two important concepts: the Neural Tangent Kernel (NTK), which captures the evolution of deep neural networks (DNNs) during training, and the Neural Collapse (NC) phenomenon, which refers to the emergence of symmetry and structure in the last-layer features of well-trained classification DNNs. We adopt the natural assumption that the empirical NTK develops a block structure aligned with the class labels, i.e., samples within the same class have stronger correlations than samples from different classes. Under this assumption, we derive the dynamics of DNNs trained with mean squared (MSE) loss and break them into interpretable phases. Moreover, we identify an invariant that captures the essence of the dynamics, and use it to prove the emergence of NC in DNNs with block-structured NTK. We provide large-scale numerical experiments on three common DNN architectures and three benchmark datasets to support our theory.

In the Fourth Industrial Revolution, wherein artificial intelligence and the automation of machines occupy a central role, the deployment of robots is indispensable. However, the manufacturing process using robots, especially in collaboration with humans, is highly intricate. In particular, modeling the friction torque in robotic joints is a longstanding problem due to the lack of a good mathematical description. This motivates the usage of data-driven methods in recent works. However, model-based and data-driven models often exhibit limitations in their ability to generalize beyond the specific dynamics they were trained on, as we demonstrate in this paper. To address this challenge, we introduce a novel approach based on residual learning, which aims to adapt an existing friction model to new dynamics using as little data as possible. We validate our approach by training a base neural network on a symmetric friction data set to learn an accurate relation between the velocity and the friction torque. Subsequently, to adapt to more complex asymmetric settings, we train a second network on a small dataset, focusing on predicting the residual of the initial network's output. By combining the output of both networks in a suitable manner, our proposed estimator outperforms the conventional model-based approach and the base neural network significantly. Furthermore, we evaluate our method on trajectories involving external loads and still observe a substantial improvement, approximately 60-70\%, over the conventional approach. Our method does not rely on data with external load during training, eliminating the need for external torque sensors. This demonstrates the generalization capability of our approach, even with a small amount of data-only 43 seconds of a robot movement-enabling adaptation to diverse scenarios based on prior knowledge about friction in different settings.

Deep neural networks have seen tremendous success over the last years. Since the training is performed on digital hardware, in this paper, we analyze what actually can be computed on current hardware platforms modeled as Turing machines, which would lead to inherent restrictions of deep learning. For this, we focus on the class of inverse problems, which, in particular, encompasses any task to reconstruct data from measurements. We prove that finite-dimensional inverse problems are not Banach-Mazur computable for small relaxation parameters. Even more, our results introduce a lower bound on the accuracy that can be obtained algorithmically.

While large language models demonstrate remarkable capabilities, they often present challenges in terms of safety, alignment with human values, and stability during training. Here, we focus on two prevalent methods used to align these models, Supervised Fine-Tuning (SFT) and Reinforcement Learning from Human Feedback (RLHF). SFT is simple and robust, powering a host of open-source models, while RLHF is a more sophisticated method used in top-tier models like ChatGPT but also suffers from instability and susceptibility to reward hacking. We propose a novel approach, Supervised Iterative Learning from Human Feedback (SuperHF), which seeks to leverage the strengths of both methods. Our hypothesis is two-fold: that the reward model used in RLHF is critical for efficient data use and model generalization and that the use of Proximal Policy Optimization (PPO) in RLHF may not be necessary and could contribute to instability issues. SuperHF replaces PPO with a simple supervised loss and a Kullback-Leibler (KL) divergence prior. It creates its own training data by repeatedly sampling a batch of model outputs and filtering them through the reward model in an online learning regime. We then break down the reward optimization problem into three components: robustly optimizing the training rewards themselves, preventing reward hacking-exploitation of the reward model that degrades model performance-as measured by a novel METEOR similarity metric, and maintaining good performance on downstream evaluations. Our experimental results show SuperHF exceeds PPO-based RLHF on the training objective, easily and favorably trades off high reward with low reward hacking, improves downstream calibration, and performs the same on our GPT-4 based qualitative evaluation scheme all the while being significantly simpler to implement, highlighting SuperHF's potential as a competitive language model alignment technique.