Fast Reconstruction of Exact Maxwell Dynamics from Sparse Data

Generating dense physical fields from sparse measurements is a fundamental question in sampling, signal processing, and many other applications. State-of-the-art approaches to this problem either rely on spatial statistics that ignore the governing physics, integrate the physics into a multiple-objective optimization process, or require examples of the complete, fully-resolved simulation state during training, which are frequently unavailable outside of synthetic benchmarks. Here, we present a novel alternative that leverages recent advances in the integration of numerical simulators with data-driven models. Namely, we propose a hybrid modeling pipeline that couples Radial Basis Function (RBF) reconstruction with a Neural Network (NN) correction and a Partial Differential Equation (PDE) solver, so that the numerical simulator itself is embedded directly in the training loop of the learned component. Notably, the NN is trained without assuming availability of examples of the fully-resolved simulation state. This is made possible by implementing the PDE solver so that it is end-to-end differentiable, allowing gradients to be backpropagated through the simulation step during training. This grey-box methodology is evaluated on three standard benchmarks from fluid mechanics, where it achieves superior results over statistical and machine-learning-based reconstruction methods.

Many multivariate dynamical systems are observed only through trajectories, leaving the mechanisms governing their joint dynamics hidden. Existing approaches can impose interpretable dynamics or learn flexible state transitions, yet the resulting interaction structure is typically either specified in advance or left implicit within the learned dynamics. We introduce MF-Net, a recurrent dynamical model that represents all variables in a shared field state and updates this state through a learned relation law. Each variable carries a field component, and these components evolve jointly through a learnable mechanical transition. Here, mechanical refers to the relation-to-motion organization of the transition, where learned relations shape state-dependent flows, field responses, and motion tendencies that move the field state forward. The resulting structure is part of the rollout itself: learned relations influence how the field moves, and the same internal quantities support both forecasting and structural readout. Across known-law interaction systems, chaotic benchmarks, real neural recordings, and ecological time series, MF-Net achieves competitive short- and medium-horizon forecasting while retaining inspectable structural readout. On the 40-dimensional Lorenz--96 testbed, MF-Net achieves an eight-step $R^2$ of $0.798\pm0.018$; across five seeds, its learned relation matrix recovers the local coupling support with a local/nonlocal strength ratio of $19.80\pm1.00$ and Precision@$K$ of $1.000\pm0.000$. MF-Net provides a structure-readable dynamical modeling framework in which learned relations are trained through forward evolution and, on real data, interpreted as functional predictive couplings under appropriate observational limits.

Predicting electromagnetic field (EMF) strength in urban environments is essential for cellular network planning but computationally expensive with physics-based simulators. We propose a multi-conditioned dense prediction framework that generates 500 500 EMF maps from building layout images and antenna configurations. Our architecture uses a High-Resolution Transformer (HRFormer) backbone with two complementary conditioning mechanisms: Feature-wise Linear Modulation (FiLM) injects scalar antenna parameters into all backbone stages, while cross-attention fuses 1-D radiation pattern tokens with spatial features at the deepest stage. We further introduce transmitter-relative spatial channels encoding distance, proximity, and bearing from the antenna, enabling coordinate-consistent test-time augmentation (TTA) that reduces test MAE by 6.3%. To address the prediction difficulty imbalance across EMF maps, we design a composite loss combining masked L1, multi-scale structural similarity (MS-SSIM), and a focal L1 term that upweights high-signal pixels, outperforming individual loss components in all metrics. Our best model achieves a test MAE of 0.0461, a 25.2% improvement over a plain UNet baseline and 31.8% over an HRFormer-only baseline.Do-