The Monte Carlo-type Neural Operator (MCNO) introduces a framework for learning solution operators of one-dimensional partial differential equations (PDEs) by directly learning the kernel function and approximating the associated integral operator using a Monte Carlo-type approach. Unlike Fourier Neural Operators (FNOs), which rely on spectral representations and assume translation-invariant kernels, MCNO makes no such assumptions. The kernel is represented as a learnable tensor acting on the latent features at sampled points, with sampling performed once uniformly at random from the discretized grid. This design enables generalization across multiple grid resolutions without relying on fixed global basis functions or repeated sampling during training, while an interpolation step maps between arbitrary input and output grids to further enhance flexibility. Experiments on standard 1D PDE benchmarks show that MCNO achieves competitive accuracy with efficient computational cost. We also provide a theoretical analysis proving that the Monte Carlo estimator yields a bounded bias and variance under mild regularity assumptions. This result holds in any spatial dimension, suggesting that MCNO may extend naturally beyond one-dimensional problems. More broadly, this work explores how Monte Carlo-type integration can be incorporated into neural operator frameworks for continuous-domain PDEs, providing a theoretically supported alternative to spectral methods (such as FNO) and to graph-based Monte Carlo approaches (such as the Graph Kernel Neural Operator, GNO).

We introduce FlowMixer, a single-layer neural architecture that leverages constrained matrix operations to model structured spatiotemporal patterns with enhanced interpretability. FlowMixer incorporates non-negative matrix mixing layers within a reversible mapping framework—applying transforms before mixing and their inverses afterward. This shape-preserving design enables a Kronecker-Koopman eigenmodes framework that bridges statistical learning with dynamical systems theory, providing interpretable spatiotemporal patterns and facilitating direct algebraic manipulation of prediction horizons without retraining. The architecture's semi-group property enables this single layer to mathematically represent any depth through composition, eliminating depth search entirely. Extensive experiments across diverse domains demonstrate FlowMixer's long-horizon forecasting capabilities while effectively modeling physical phenomena such as chaotic attractors and turbulent flows. Our results achieve performance matching state-of-the-art methods while offering superior interpretability through directly extractable eigenmodes. This work suggests that architectural constraints can simultaneously maintain competitive performance and enhance mathematical interpretability in neural forecasting systems.

We consider the role of non-localities in speed-density data used to fit fundamental diagrams from vehicle trajectories. We demonstrate that the use of anticipated densities results in a clear classification of speed-density data into stationary and non-stationary points, namely, acceleration and deceleration regimes and their separating boundary. The separating boundary represents a locus of stationary traffic states, i.e., the fundamental diagram. To fit fundamental diagrams, we develop an enhanced cross entropy minimization method that honors equilibrium traffic physics. We illustrate the effectiveness of our proposed approach by comparing it with the traditional approach that uses local speed-density states and least squares estimation. Our experiments show that the separating boundary in our approach is invariant to varying trajectory samples within the same spatio-temporal region, providing further evidence that the separating boundary is indeed a locus of stationary traffic states.

The rapid development and deployment of vehicle technologies offer opportunities to re-think the way traffic is managed. This article capitalizes on vehicle connectivity and proposes an economic instrument and corresponding cooperative framework for allocating priority at intersections. The framework is compatible with a variety of existing intersection control approaches. Similar to free markets, our framework allows vehicles to trade their time based on their (disclosed) value of time. We design the framework based on transferable utility games, where winners (time buyers) pay losers (time sellers) in each game. We conduct simulation experiments of both isolated intersections and an arterial setting. The results show that the proposed approach benefits the majority of users when compared to other mechanisms both ones that employ an economic instrument and ones that do not. We also show that it drives travelers to estimate their value of time correctly, and it naturally dissuades travelers from attempting to cheat.

We propose a kinematic wave-based Deep Convolutional Neural Network (Deep CNN) to estimate high-resolution traffic speed fields from sparse probe vehicle trajectories. We introduce two key approaches that allow us to incorporate kinematic wave theory principles to improve the robustness of existing learning-based estimation methods. First, we propose an anisotropic traffic kernel for the Deep CNN. The anisotropic kernel explicitly accounts for space-time correlations in macroscopic traffic and effectively reduces the number of trainable parameters in the Deep CNN model. Second, we propose to use simulated data for training the Deep CNN. Using a targeted simulated data for training provides an implicit way to impose desirable traffic physical features on the learning model. In the experiments, we highlight the benefits of using anisotropic kernels and evaluate the transferability of the trained model to real-world traffic using the Next Generation Simulation (NGSIM) and the German Highway Drone (HighD) datasets. The results demonstrate that anisotropic kernels significantly reduce model complexity and model over-fitting, and improve the physical correctness of the estimated speed fields. We find that model complexity scales linearly with problem size for anisotropic kernels compared to quadratic scaling for isotropic kernels. Furthermore, evaluation on real-world datasets shows acceptable performance, which establishes that simulation-based training is a viable surrogate to learning from real-world data. Finally, a comparison with standard estimation techniques shows the superior estimation accuracy of the proposed method.

Recent work has shown that the introduction of autonomous vehicles (AVs) in traffic could help reduce traffic jams. Deep reinforcement learning methods demonstrate good performance in complex control problems, including autonomous vehicle control, and have been used in state-of-the-art AV controllers. However, deep neural networks (DNNs) render automated driving vulnerable to machine learning-based attacks. In this work, we explore the backdooring/trojanning of DRL-based AV controllers. We develop a trigger design methodology that is based on well-established principles of traffic physics. The malicious actions include vehicle deceleration and acceleration to cause stop-and-go traffic waves to emerge (congestion attacks) or AV acceleration resulting in the AV crashing into the vehicle in front (insurance attack). We test our attack on single-lane and two-lane circuits. Our experimental results show that the backdoored model does not compromise normal operation performance, with the maximum decrease in cumulative rewards being 1%. Still, it can be maliciously activated to cause a crash or congestion when the corresponding triggers appear.