Arghya Pathak

Aritra Das, Yashas Shende, Muskaan Chugh et al.

Magnetic-anomaly navigation, leveraging small-scale variations in the Earth's magnetic field, is a promising alternative when GPS is unavailable or compromised. Airborne systems face a key challenge in extracting geomagnetic field data: the aircraft itself induces magnetic noise. Although the classical Tolles-Lawson model addresses this, it inadequately handles stochastically corrupted magnetic data required for navigation. To handle stochastic noise, we propose using two physics-based constraints: divergence-free vector fields and E(3)-equivariance. These ensure the learned magnetic field obeys Maxwell's equation and that outputs transform correctly with sensor position and orientation. The divergence-free constraint is implemented by training a neural network to output a vector potential A, with the magnetic field defined as its curl. For E(3)-equivariance, we use tensor products of geometric tensors represented via spherical harmonics with known rotational transformations. Enforcing physical consistency and restricting the admissible function space acts as an implicit regularizer that improves spatiotemporal performance. We present ablation studies evaluating each constraint alone and jointly across CNNs, MLPs, LTCs, and Contiformers. Continuous-time dynamics and long-term memory are critical for modelling magnetic time series; the Contiformer, which provides both, outperforms existing methods. To mitigate data scarcity, we generate synthetic datasets using the World Magnetic Model (WMM) and time-series conditional GANs, producing realistic, temporally consistent magnetic sequences across varied trajectories and environments. Experiments show that embedding these constraints significantly improves predictive accuracy and physical plausibility, outperforming classical and unconstrained deep learning approaches. Acknowledgement: This work was done in collaboration with Dirac Labs.

Yashas Shende, Aritra Das, Reva Laxmi Chauhan et al.

Transformers excel at time series modelling through attention mechanisms that capture long-term temporal patterns. However, they assume uniform time intervals and therefore struggle with irregular time series. Neural Ordinary Differential Equations (NODEs) effectively handle irregular time series by modelling hidden states as continuously evolving trajectories. ContiFormers arxiv:2402.10635 combine NODEs with Transformers, but inherit the computational bottleneck of the former by using heavy numerical solvers. This bottleneck can be removed by using a closed-form solution for the given dynamical system - but this is known to be intractable in general! We obviate this by replacing NODEs with a novel linear damped harmonic oscillator analogy - which has a known closed-form solution. We model keys and values as damped, driven oscillators and expand the query in a sinusoidal basis up to a suitable number of modes. This analogy naturally captures the query-key coupling that is fundamental to any transformer architecture by modelling attention as a resonance phenomenon. Our closed-form solution eliminates the computational overhead of numerical ODE solvers while preserving expressivity. We prove that this oscillator-based parameterisation maintains the universal approximation property of continuous-time attention; specifically, any discrete attention matrix realisable by ContiFormer's continuous keys can be approximated arbitrarily well by our fixed oscillator modes. Our approach delivers both theoretical guarantees and scalability, achieving state-of-the-art performance on irregular time series benchmarks while being orders of magnitude faster.