In conditioned generative models of physical systems, the densities at different values of the conditioning parameters are often related through symplectic transformations. For stable linear Hamiltonian systems, there exists a symplectic transformation to "normal coordinates" such that the dynamics is reduced to rotations in each phase-space plane. This picture extends to the nonlinear case where, away from resonances, the Birkhoff normal form again consists of rotations in phase space, now with amplitude dependence. In this work, we introduce an invertible symplectic flow layer consisting of amplitude-dependent rotations in learned normal coordinates. We derive the condition under which the nonlinear rotation layer is symplectic and detail an architecture that guarantees it. Experiments are carried out on three conditional generative modeling problems: a synthetic chain of coupled nonlinear oscillators and two problems from accelerator physics. We show empirically that normalizing flows with the new layers achieve competitive or superior negative log-likelihoods compared to baseline flows, with gains that grow with the strength of the nonlinearity on the synthetic system.
Large-scale scientific facilities, such as particle accelerators, are complex systems composed of thousands of interacting components that must operate collectively to support diagnostics and control, and ensure safe, reliable operation. These facilities rely on extensive sensor networks that generate large volumes of heterogeneous, incomplete, and noisy telemetry, often sampled at different rates across subsystems. In this paper, we adopt the foundation-model paradigm to learn general-purpose sensor representations from historical accelerator data through self-supervised pretraining. We introduce SensOFormer, a denoising masked transformer with a Perceiver-style encoder-decoder architecture designed to handle variable sensor sets, accommodate heterogeneous sampling rates, and learn robust representations from noisy measurements. The model is pretrained on multiple particle-accelerator datasets and transferred to downstream tasks, including missing-data imputation, anomaly detection, cavity identification, and fault identification. Through extensive evaluations and comparisons, we demonstrate that a single self-supervised model can learn reusable representations for diverse operational tasks in large-scale scientific facilities. Project page: https://mahindrautela.github.io/sensoformer.
The electron beams of a laser-plasma accelerator depend non-linearly on the laser and gas settings. This dependence is explored using particle-in-cell (PIC) simulations, and machine-learning models trained on these simulations predict the beam from the settings. Here we address the reverse problem: given a measured beam, which settings produced it? We infer three of them: the gas pressure $p\_1$, the normalised vector potential $a\_0$ and the nitrogen concentration $c\_{N\_2}$. The solution is not guaranteed to be unique. We train an ensemble of mixture density networks that, for each beam, output several possible sets of settings with their probabilities and uncertainties, and test them on simulated beams with realistic measurement noise. On average, the predictions are accurate ($R^2 = 0.91$), yet for about two-thirds of the beams the inferred distribution has two competing solutions. These share nearly the same value of $a\_0^2 p\_1$, which controls how strongly the laser focuses itself in the plasma and, in earlier PALLAS studies, when electrons start to be injected. Keeping this quantity fixed gives beams that are practically indistinguishable with realistic diagnostics. A forward model used as a check removes unphysical solutions but cannot choose between the two plausible ones. We then test two ways to resolve the ambiguity. Adding the full electron energy spectrum improves the mean prediction ($R^2 = 0.94$), mostly through $a\_0$. It changes the weights of the two solutions but does not move them. A second measurement at a deliberately different plasma density is more effective. In paired PIC simulations, it reduces the error on $a\_0$ by a factor of 2.6, whereas repeating the measurement at the same density brings almost no gain.