Where Do We Stand with Implicit Neural Representations? A Technical and Performance Survey

Implicit Neural Representations (INRs) have emerged as a paradigm in knowledge representation, offering exceptional flexibility and performance across a diverse range of applications. INRs leverage multilayer perceptrons (MLPs) to model data as continuous implicit functions, providing critical advantages such as resolution independence, memory efficiency, and generalisation beyond discretised data structures. Their ability to solve complex inverse problems makes them particularly effective for tasks including audio reconstruction, image representation, 3D object reconstruction, and high-dimensional data synthesis. This survey provides a comprehensive review of state-of-the-art INR methods, introducing a clear taxonomy that categorises them into four key areas: activation functions, position encoding, combined strategies, and network structure optimisation. We rigorously analyse their critical properties, such as full differentiability, smoothness, compactness, and adaptability to varying resolutions while also examining their strengths and limitations in addressing locality biases and capturing fine details. Our experimental comparison offers new insights into the trade-offs between different approaches, showcasing the capabilities and challenges of the latest INR techniques across various tasks. In addition to identifying areas where current methods excel, we highlight key limitations and potential avenues for improvement, such as developing more expressive activation functions, enhancing positional encoding mechanisms, and improving scalability for complex, high-dimensional data. This survey serves as a roadmap for researchers, offering practical guidance for future exploration in the field of INRs. We aim to foster new methodologies by outlining promising research directions for INRs and applications.

On the Utilization of Unique Node Identifiers in Graph Neural Networks

Graph neural networks have inherent representational limitations due to their message-passing structure. Recent work has suggested that these limitations can be overcome by using unique node identifiers (UIDs). Here we argue that despite the advantages of UIDs, one of their disadvantages is that they lose the desirable property of permutation-equivariance. We thus propose to focus on UID models that are permutation-equivariant, and present theoretical arguments for their advantages. Motivated by this, we propose a method to regularize UID models towards permutation equivariance, via a contrastive loss. We empirically demonstrate that our approach improves generalization and extrapolation abilities while providing faster training convergence. On the recent BREC expressiveness benchmark, our proposed method achieves state-of-the-art performance compared to other random-based approaches.

Parameter choices in HaarPSI for IQA with medical images

When developing machine learning models, image quality assessment (IQA) measures are a crucial component for evaluation. However, commonly used IQA measures have been primarily developed and optimized for natural images. In many specialized settings, such as medical images, this poses an often-overlooked problem regarding suitability. In previous studies, the IQA measure HaarPSI showed promising behavior for natural and medical images. HaarPSI is based on Haar wavelet representations and the framework allows optimization of two parameters. So far, these parameters have been aligned for natural images. Here, we optimize these parameters for two annotated medical data sets, a photoacoustic and a chest X-Ray data set. We observe that they are more sensitive to the parameter choices than the employed natural images, and on the other hand both medical data sets lead to similar parameter values when optimized. We denote the optimized setting, which improves the performance for the medical images notably, by HaarPSI$_{MED}$. The results suggest that adapting common IQA measures within their frameworks for medical images can provide a valuable, generalizable addition to the employment of more specific task-based measures.

Hamiltonian Matching for Symplectic Neural Integrators

Hamilton's equations of motion form a fundamental framework in various branches of physics, including astronomy, quantum mechanics, particle physics, and climate science. Classical numerical solvers are typically employed to compute the time evolution of these systems. However, when the system spans multiple spatial and temporal scales numerical errors can accumulate, leading to reduced accuracy. To address the challenges of evolving such systems over long timescales, we propose SympFlow, a novel neural network-based symplectic integrator, which is the composition of a sequence of exact flow maps of parametrised time-dependent Hamiltonian functions. This architecture allows for a backward error analysis: we can identify an underlying Hamiltonian function of the architecture and use it to define a Hamiltonian matching objective function, which we use for training. In numerical experiments, we show that SympFlow exhibits promising results, with qualitative energy conservation behaviour similar to that of time-stepping symplectic integrators.

Pullback Flow Matching on Data Manifolds

We propose Pullback Flow Matching (PFM), a novel framework for generative modeling on data manifolds. Unlike existing methods that assume or learn restrictive closed-form manifold mappings for training Riemannian Flow Matching (RFM) models, PFM leverages pullback geometry and isometric learning to preserve the underlying manifold's geometry while enabling efficient generation and precise interpolation in latent space. This approach not only facilitates closed-form mappings on the data manifold but also allows for designable latent spaces, using assumed metrics on both data and latent manifolds. By enhancing isometric learning through Neural ODEs and proposing a scalable training objective, we achieve a latent space more suitable for interpolation, leading to improved manifold learning and generative performance. We demonstrate PFM's effectiveness through applications in synthetic data, protein dynamics and protein sequence data, generating novel proteins with specific properties. This method shows strong potential for drug discovery and materials science, where generating novel samples with specific properties is of great interest.

Lie Algebra Canonicalization: Equivariant Neural Operators under arbitrary Lie Groups

The quest for robust and generalizable machine learning models has driven recent interest in exploiting symmetries through equivariant neural networks. In the context of PDE solvers, recent works have shown that Lie point symmetries can be a useful inductive bias for Physics-Informed Neural Networks (PINNs) through data and loss augmentation. Despite this, directly enforcing equivariance within the model architecture for these problems remains elusive. This is because many PDEs admit non-compact symmetry groups, oftentimes not studied beyond their infinitesimal generators, making them incompatible with most existing equivariant architectures. In this work, we propose Lie aLgebrA Canonicalization (LieLAC), a novel approach that exploits only the action of infinitesimal generators of the symmetry group, circumventing the need for knowledge of the full group structure. To achieve this, we address existing theoretical issues in the canonicalization literature, establishing connections with frame averaging in the case of continuous non-compact groups. Operating within the framework of canonicalization, LieLAC can easily be integrated with unconstrained pre-trained models, transforming inputs to a canonical form before feeding them into the existing model, effectively aligning the input for model inference according to allowed symmetries. LieLAC utilizes standard Lie group descent schemes, achieving equivariance in pre-trained models. Finally, we showcase LieLAC's efficacy on tasks of invariant image classification and Lie point symmetry equivariant neural PDE solvers using pre-trained models.

Mamba Neural Operator: Who Wins? Transformers vs. State-Space Models for PDEs

Partial differential equations (PDEs) are widely used to model complex physical systems, but solving them efficiently remains a significant challenge. Recently, Transformers have emerged as the preferred architecture for PDEs due to their ability to capture intricate dependencies. However, they struggle with representing continuous dynamics and long-range interactions. To overcome these limitations, we introduce the Mamba Neural Operator (MNO), a novel framework that enhances neural operator-based techniques for solving PDEs. MNO establishes a formal theoretical connection between structured state-space models (SSMs) and neural operators, offering a unified structure that can adapt to diverse architectures, including Transformer-based models. By leveraging the structured design of SSMs, MNO captures long-range dependencies and continuous dynamics more effectively than traditional Transformers. Through extensive analysis, we show that MNO significantly boosts the expressive power and accuracy of neural operators, making it not just a complement but a superior framework for PDE-related tasks, bridging the gap between efficient representation and accurate solution approximation.

Bellman Diffusion: Generative Modeling as Learning a Linear Operator in the Distribution Space

Deep Generative Models (DGMs), including Energy-Based Models (EBMs) and Score-based Generative Models (SGMs), have advanced high-fidelity data generation and complex continuous distribution approximation. However, their application in Markov Decision Processes (MDPs), particularly in distributional Reinforcement Learning (RL), remains underexplored, with conventional histogram-based methods dominating the field. This paper rigorously highlights that this application gap is caused by the nonlinearity of modern DGMs, which conflicts with the linearity required by the Bellman equation in MDPs. For instance, EBMs involve nonlinear operations such as exponentiating energy functions and normalizing constants. To address this, we introduce Bellman Diffusion, a novel DGM framework that maintains linearity in MDPs through gradient and scalar field modeling. With divergence-based training techniques to optimize neural network proxies and a new type of stochastic differential equation (SDE) for sampling, Bellman Diffusion is guaranteed to converge to the target distribution. Our empirical results show that Bellman Diffusion achieves accurate field estimations and is a capable image generator, converging 1.5x faster than the traditional histogram-based baseline in distributional RL tasks. This work enables the effective integration of DGMs into MDP applications, unlocking new avenues for advanced decision-making frameworks.

Score-based pullback Riemannian geometry

Data-driven Riemannian geometry has emerged as a powerful tool for interpretable representation learning, offering improved efficiency in downstream tasks. Moving forward, it is crucial to balance cheap manifold mappings with efficient training algorithms. In this work, we integrate concepts from pullback Riemannian geometry and generative models to propose a framework for data-driven Riemannian geometry that is scalable in both geometry and learning: score-based pullback Riemannian geometry. Focusing on unimodal distributions as a first step, we propose a score-based Riemannian structure with closed-form geodesics that pass through the data probability density. With this structure, we construct a Riemannian autoencoder (RAE) with error bounds for discovering the correct data manifold dimension. This framework can naturally be used with anisotropic normalizing flows by adopting isometry regularization during training. Through numerical experiments on various datasets, we demonstrate that our framework not only produces high-quality geodesics through the data support, but also reliably estimates the intrinsic dimension of the data manifold and provides a global chart of the manifold, even in high-dimensional ambient spaces.

Learning Regularization for Graph Inverse Problems

In recent years, Graph Neural Networks (GNNs) have been utilized for various applications ranging from drug discovery to network design and social networks. In many applications, it is impossible to observe some properties of the graph directly; instead, noisy and indirect measurements of these properties are available. These scenarios are coined as Graph Inverse Problems (GRIP). In this work, we introduce a framework leveraging GNNs to solve GRIPs. The framework is based on a combination of likelihood and prior terms, which are used to find a solution that fits the data while adhering to learned prior information. Specifically, we propose to combine recent deep learning techniques that were developed for inverse problems, together with GNN architectures, to formulate and solve GRIP. We study our approach on a number of representative problems that demonstrate the effectiveness of the framework.