Research Projects

This project proposes a novel probabilistic, unified and physics-informed framework for solving both the forward and inverse problems associated with the Radiative Transport Equation (RTE), with a focus on real-time reconstruction of electromagnetic (EM) sources from noisy boundary measurements. The innovation lies in combining the mathematical rigor of optimal transport with the flexibility of deep learning, enabling robust and efficient inference even in the presence of sparse data and singular source terms such as Dirac deltas. Traditional RTE solvers (FEM, deterministic PINNs) often struggle to handle the ill-posedness, stiffness, and singular sources inherent in practical applications such as EM source detection. We leverage the Jordan–Kinderlehrer–Otto (JKO) variational framework to interpret the evolution of probability measures as a Wasserstein gradient flow. This geometric, i.e. Riemannian, structure is used to efficaciously guide the training of neural networks that approximate solutions to the RTE. Depending on whether we are solving the forward or the inverse problem, the squared residual of the RTE or that of a nonlinear filtering equation (such as the Kushner–Stratonovich or Zakai equation) is treated as a potential energy functional in the JKO flow. This is combined with Wasserstein transport cost and entropy regularization to form a loss functional that balances data fidelity, physical consistency, and statistical smoothness. Neural ordinary or stochastic differential equations serve as pushforward maps from Gaussian base distributions, enabling efficient probabilistic representations of the evolving solution. To account for multiple loss components—e.g., residuals, entropy, transport cost—a game-theoretic multi-objective optimization strategy, such as Nash-MTL or MGDA, is employed. This avoids the trap of manual loss weighting and ensures balanced training across competing objectives. The forward solver simulates EM propagation through scattering media under known sources and boundary conditions. The inverse solver reconstructs source location and intensity from partial and noisy boundary observations. Beyond the training of the networks via simulations, a lab-based demonstration involving laser sources and controlled scattering environments will validate the methodology. Key deliverables include a modular software toolkit for forward and inverse solvers, validated against synthetic and laboratory data, and yearly reports culminating in deployable algorithms for real-time EM source detection. The method’s modularity enables adaptation to broader applications such as medical imaging, atmospheric sensing, and non-destructive testing. In defence contexts, this approach supports strategic EM surveillance, passive detection, and threat localization, providing real-time intelligence in complex environments. By integrating modern machine learning with measure-theoretic physical insights and optimal transport, the project offers a scalable and theoretically grounded approach to challenging inverse problems. The immediate applications are in defence and sensing, such as electronic warfare, counter-surveillance, and situational awareness. These include passive detection of covert or unauthorized EM emitters and interference analyses in communication and radar systems. However, the universality of the proposed framework for inverse problems can naturally accommodate applications across a swathe of areas of engineering interest, including medical imaging (e.g., near-infrared tomography), environmental monitoring (e.g., pollutant source tracking), non-destructive testing, structural health assessment and atmospheric radiation modelling.