Neural Information Processing Systems http://nips.cc/

This work studies operators mapping vector and scalar fields defined over a manifold $\mathcal{M}$, and which commute with its group of diffeomorphisms $\text{Diff}(\mathcal{M})$. We prove that in the case of scalar fields $L^p_\omega(\mathcal{M,\mathbb{R}})$, those operators correspond to point-wise non-linearities, recovering and extending known results on $\mathbb{R}^d$. In the context of Neural Networks defined over $\mathcal{M}$, it indicates that point-wise non-linear operators are the only universal family that commutes with any group of symmetries, and justifies their systematic use in combination with dedicated linear operators commuting with specific symmetries. In the case of vector fields $L^p_\omega(\mathcal{M},T\mathcal{M})$, we show that those operators are solely the scalar multiplication. It indicates that $\text{Diff}(\mathcal{M})$ is too rich and that there is no universal class of non-linear operators to motivate the design of Neural Networks over the symmetries of $\mathcal{M}$.

Scientific simulations and experimental measurements produce vast amounts of spatio-temporal data, yet extracting meaningful insights remains challenging due to high dimensionality, complex structures, and missing information. Traditional analysis methods often struggle with these issues, motivating the need for more robust, data-driven approaches. This dissertation explores deep learning methodologies to improve the analysis and visualization of spatio-temporal scientific ensembles, focusing on dimensionality reduction, flow estimation, and temporal interpolation. First, we address high-dimensional data representation through autoencoder-based dimensionality reduction for scientific ensembles. We evaluate the stability of projection metrics under partial labeling and introduce a Pareto-efficient selection strategy to identify optimal autoencoder variants, ensuring expressive and reliable low-dimensional embeddings. Next, we present FLINT, a deep learning model for high-quality flow estimation and temporal interpolation in both flow-supervised and flow-unsupervised settings. FLINT reconstructs missing velocity fields and generates high-fidelity temporal interpolants for scalar fields across 2D+time and 3D+time ensembles without domain-specific assumptions or extensive finetuning. To further improve adaptability and generalization, we introduce HyperFLINT, a hypernetwork-based approach that conditions on simulation parameters to estimate flow fields and interpolate scalar data. This parameter-aware adaptation yields more accurate reconstructions across diverse scientific domains, even with sparse or incomplete data. Overall, this dissertation advances deep learning techniques for scientific visualization, providing scalable, adaptable, and high-quality solutions for interpreting complex spatio-temporal ensembles.

Existing curved-layer-based process planning methods for multi-axis manufacturing address collisions only indirectly and generate toolpaths in a post-processing step, leaving toolpath geometry uncontrolled during optimization. We present an implicit neural field-based framework for multi-axis process planning that overcomes these limitations by embedding both layer generation and toolpath design within a single differentiable pipeline. Using sinusoidally activated neural networks to represent layers and toolpaths as implicit fields, our method enables direct evaluation of field values and derivatives at any spatial point, thereby allowing explicit collision avoidance and joint optimization of manufacturing layers and toolpaths. We further investigate how network hyperparameters and objective definitions influence singularity behavior and topology transitions, offering built-in mechanisms for regularization and stability control. The proposed approach is demonstrated on examples in both additive and subtractive manufacturing, validating its generality and effectiveness.

This paper presents a neural scheme for the topology-aware interpolation of time-varying scalar fields. Given a time-varying sequence of persistence diagrams, along with a sparse temporal sampling of the corresponding scalar fields, denoted as keyframes, our interpolation approach aims at "inverting" the non-keyframe diagrams to produce plausible estimations of the corresponding, missing data. For this, we rely on a neural architecture which learns the relation from a time value to the corresponding scalar field, based on the keyframe examples, and reliably extends this relation to the non-keyframe time steps. We show how augmenting this architecture with specific topological losses exploiting the input diagrams both improves the geometrical and topological reconstruction of the non-keyframe time steps. At query time, given an input time value for which an interpolation is desired, our approach instantaneously produces an output, via a single propagation of the time input through the network. Experiments interpolating 2D and 3D time-varying datasets show our approach superiority, both in terms of data and topological fitting, with regard to reference interpolation schemes. Our implementation is available at this GitHub link : https://github.com/MohamedKISSI/Topology-Aware-Neural-Interpolation-of-Scalar-Fields.git.

We present a convolutional network that is equivariant to rigid body motions.

Machine learning models for molecular property prediction generally rely on representations -- such as SMILES strings and molecular graphs -- that overlook the surface-local phenomena driving intermolecular behavior. 3D-based approaches often reduce surface detail or require computationally expensive SE(3)-equivariant architectures to manage spatial variance. To overcome these limitations, this work introduces AMPTCR (Aligned Manifold Property and Topology Cloud Representation), a molecular surface representation that combines local quantum-derived scalar fields and custom topological descriptors within an aligned point cloud format. Each surface point includes a chemically meaningful scalar, geodesically derived topology vectors, and coordinates transformed into a canonical reference frame, enabling efficient learning with conventional SE(3)-sensitive architectures. AMPTCR is evaluated using a DGCNN framework on two tasks: molecular weight and bacterial growth inhibition. For molecular weight, results confirm that AMPTCR encodes physically meaningful data, with a validation R^2 of 0.87. In the bacterial inhibition task, AMPTCR enables both classification and direct regression of E. coli inhibition values using Dual Fukui functions as the electronic descriptor and Morgan Fingerprints as auxiliary data, achieving an ROC AUC of 0.912 on the classification task, and an R^2 of 0.54 on the regression task. These results help demonstrate that AMPTCR offers a compact, expressive, and architecture-agnostic representation for modeling surface-mediated molecular properties.

This work addresses the critical need to explicitly incorporate ensemble parameters into the learning process, as traditional methods often neglect these, limiting their ability to adapt to diverse simulation settings and provide meaningful insights into the data dynamics. HyperFLINT introduces a hypernetwork to account for simulation parameters, enabling it to generate accurate interpolations and flow fields for each timestep by dynamically adapting to varying conditions, thereby outperforming existing parameter-agnostic approaches. The architecture features modular neural blocks with convolutional and deconvolutional layers, supported by a hypernetwork that generates weights for the main network, allowing the model to better capture intricate simulation dynamics. A series of experiments demonstrates HyperFLINT's significantly improved performance in flow field estimation and temporal interpolation, as well as its potential in enabling parameter space exploration, offering valuable insights into complex scientific ensembles.