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Flow visualization inherently depends on a reference frame or observer, relative to which velocities are measured. Recent research has addressed the explicit computation of optimal observers. However, although all techniques share many similarities, the differences can be significant, and no method fulfills all desirable criteria. More accurate methods can be slow, whereas faster approaches can be less accurate, which means that the observers are less optimal. In this paper, we present a novel generic variational framework that combines the best of both worlds, and offers better trade-offs between accuracy and computational efficiency. We build on ideas from Lagrangian and Hamiltonian mechanics to unify previous methods, while at the same time enabling more flexibility and novel algorithmic features. In this way, our framework can also serve as a principled foundation for analysis, comparison, and future advances in this area. We first formulate the observer optimization problem in terms of Lagrangian functions, where an optimal solution must solve the second-order Euler-Lagrange equation. We then expand toward a Hamiltonian perspective, which reduces the second-order equation to first-order. However, we use a hybrid Lagrangian-Hamiltonian approach to combine the advantages of each. By defining Hamiltonian vector fields in an observer phase space, optimal observers simply become integral curves. Our hybrid method computes these vector fields directly from the Lagrangian representing the cost to minimize. Finally, we show how this foundation also enables direct insights into flow field properties, by defining the novel notion of intrinsically steady flows using the Hamiltonian function.

Visualizing and verifying unsteady 3D vortex structures remains a fundamental challenge in flow visualization. While recent advancements enable the extraction of objective Lagrangian vortex corelines, visually verifying whether these candidate curves accurately represent the true center of swirling motion remains difficult. Standard visualization techniques, such as instantaneous streamlines, lack Galilean invariance and are highly dependent on the chosen frame of reference, often yielding misleading depictions. Conversely, while Lagrangian pathlines offer a physical ground truth, the local rotational component of particle motion is typically occluded by the convective motion of the flow. To address this, we propose a novel visualization technique that associates objective space-time pathlines with evolving Lagrangian corelines. Our method applies a space deformation induced by the deformation of the coreline that guarantees that no rotation component around the core's tangent is introduced. By applying this deformation to the tubular neighborhood of the coreline, we transform pathlines into the vortex-centric reference frame, effectively decoupling the local swirl of particles from the global drift and deformation. A primary application of this technique is the visual evaluation of vortex coreline candidates. By explicitly visualizing inversely transformed pathlines in context with the coreline curves, we directly reveal the swirling motion of the surrounding particles. Because this swirling motion is the physical ground truth for validating a vortex coreline, our method provides the first pathline-based visual assessment of candidate curves. We demonstrate our approach on multiple unsteady 3D flow datasets and methods for coreline extraction, revealing vortex-relative particle dynamics that are otherwise hidden by bulk transport.

Surrogate models are essential for efficient exploration of large-scale ensemble simulations. Implicit neural representations (INRs) provide a compact and continuous framework for modeling spatially structured data, but they often struggle with learning complex localized structures within scientific fields. Recent INR-based surrogates address this by augmenting INRs with explicit feature structures, but at the cost of flexibility and substantial memory overhead. In this paper, we present Feature-Adaptive INR (FA-INR), an adaptive INR-based surrogate model for high-fidelity and interpretable exploration of ensemble simulations. Instead of relying on structured feature representations, FA-INR leverages cross-attention over a learnable key-value memory bank to allocate model capacity adaptively based on the data characteristics. To further improve scalability, we introduce a coordinate-guided mixture of experts (MoE) framework that enhances both efficiency and specialization of feature representations. More importantly, the learned experts produce an interpretable partition over the simulation domain, enabling scientists to identify complex structures and perform localized parameter-space exploration. Beyond quantitative and qualitative evaluations, we also demonstrate that our learned expert specialization can reveal meaningful scientific insights and support localized sensitivity analysis.

To study complex real-world phenomena using computer simulations, scientists often rely on ensemble datasets generated from multiple simulation runs with varying parameter configurations. This process can produce ensemble datasets with many members, making traditional data analysis pipelines impractical due to limited I/O bandwidth and disk capacity. Distribution-based data representations have been proposed as a promising solution. Processing data in situ to generate compact distribution-based representations not only alleviates the challenges of limited I/O bandwidth and disk capacity but also enables uncertainty quantification, thus mitigating the risk of misinterpretation. Nevertheless, distribution-based methods inherently sacrifice spatial information of data samples within the distribution, potentially reducing precision in the data analysis pipeline. To address this issue, we introduce a deep learning model to reconstruct data volume from the distribution representation. Instead of using a model that predicts a data block directly from its distribution representation, we propose a deep learning model based on the Sinkhorn operator and Gumbel trick that learns to map samples drawn from a distribution to spatial locations within the block. The deep learning model can support high-quality downstream data analysis and visualization, provide point-wise uncertainty quantification, and guarantee the distribution of the reconstructed data block follows the block’s distribution representation.