VN-EGNN: E(3)-Equivariant Graph Neural Networks with Virtual Nodes Enhance Protein Binding Site Identification

Being able to identify regions within or around proteins, to which ligands can potentially bind, is an essential step to develop new drugs. Binding site identification methods can now profit from the availability of large amounts of 3D structures in protein structure databases or from AlphaFold predictions. Current binding site identification methods heavily rely on graph neural networks (GNNs), usually designed to output E(3)-equivariant predictions. Such methods turned out to be very beneficial for physics-related tasks like binding energy or motion trajectory prediction. However, the performance of GNNs at binding site identification is still limited potentially due to the lack of dedicated nodes that model hidden geometric entities, such as binding pockets. In this work, we extend E(n)-Equivariant Graph Neural Networks (EGNNs) by adding virtual nodes and applying an extended message passing scheme. The virtual nodes in these graphs are dedicated quantities to learn representations of binding sites, which leads to improved predictive performance. In our experiments, we show that our proposed method VN-EGNN sets a new state-of-the-art at locating binding site centers on COACH420, HOLO4K and PDBbind2020.

JAX-SPH: A Differentiable Smoothed Particle Hydrodynamics Framework

Particle-based fluid simulations have emerged as a powerful tool for solving the Navier-Stokes equations, especially in cases that include intricate physics and free surfaces. The recent addition of machine learning methods to the toolbox for solving such problems is pushing the boundary of the quality vs. speed tradeoff of such numerical simulations. In this work, we lead the way to Lagrangian fluid simulators compatible with deep learning frameworks, and propose JAX-SPH - a Smoothed Particle Hydrodynamics (SPH) framework implemented in JAX. JAX-SPH builds on the code for dataset generation from the LagrangeBench project (Toshev et al., 2023) and extends this code in multiple ways: (a) integration of further key SPH algorithms, (b) restructuring the code toward a Python library, (c) verification of the gradients through the solver, and (d) demonstration of the utility of the gradients for solving inverse problems as well as a Solver-in-the-Loop application. Our code is available at https://github.com/tumaer/jax-sph.

GNN-VPA: A Variance-Preserving Aggregation Strategy for Graph Neural Networks

Graph neural networks (GNNs), and especially message-passing neural networks, excel in various domains such as physics, drug discovery, and molecular modeling. The expressivity of GNNs with respect to their ability to discriminate non-isomorphic graphs critically depends on the functions employed for message aggregation and graph-level readout. By applying signal propagation theory, we propose a variance-preserving aggregation function (VPA) that maintains expressivity, but yields improved forward and backward dynamics. Experiments demonstrate that VPA leads to increased predictive performance for popular GNN architectures as well as improved learning dynamics. Our results could pave the way towards normalizer-free or self-normalizing GNNs.

Clifford-Steerable Convolutional Neural Networks

We present Clifford-Steerable Convolutional Neural Networks (CS-CNNs), a novel class of $\mathrm{E}(p, q)$-equivariant CNNs. CS-CNNs process multivector fields on pseudo-Euclidean spaces $\mathbb{R}^{p,q}$. They cover, for instance, $\mathrm{E}(3)$-equivariance on $\mathbb{R}^3$ and Poincar\'e-equivariance on Minkowski spacetime $\mathbb{R}^{1,3}$. Our approach is based on an implicit parametrization of $\mathrm{O}(p,q)$-steerable kernels via Clifford group equivariant neural networks. We significantly and consistently outperform baseline methods on fluid dynamics as well as relativistic electrodynamics forecasting tasks.

Geometry-Informed Neural Networks

We introduce the concept of geometry-informed neural networks (GINNs), which encompass (i) learning under geometric constraints, (ii) neural fields as a suitable representation, and (iii) generating diverse solutions to under-determined systems often encountered in geometric tasks. Notably, the GINN formulation does not require training data, and as such can be considered generative modeling driven purely by constraints. We add an explicit diversity loss to mitigate mode collapse. We consider several constraints, in particular, the connectedness of components which we convert to a differentiable loss through Morse theory. Experimentally, we demonstrate the efficacy of the GINN learning paradigm across a range of two and three-dimensional scenarios with increasing levels of complexity.

Universal Physics Transformers

Deep neural network based surrogates for partial differential equations have recently gained increased interest. However, akin to their numerical counterparts, different techniques are used across applications, even if the underlying dynamics of the systems are similar. A prominent example is the Lagrangian and Eulerian specification in computational fluid dynamics, posing a challenge for neural networks to effectively model particle- as opposed to grid-based dynamics. We introduce Universal Physics Transformers (UPTs), a novel learning paradigm which models a wide range of spatio-temporal problems - both for Lagrangian and Eulerian discretization schemes. UPTs operate without grid- or particle-based latent structures, enabling flexibility across meshes and particles. UPTs efficiently propagate dynamics in the latent space, emphasized by inverse encoding and decoding techniques. Finally, UPTs allow for queries of the latent space representation at any point in space-time. We demonstrate the efficacy of UPTs in mesh-based fluid simulations, steady-state Reynolds averaged Navier-Stokes simulations, and Lagrangian-based dynamics. Project page: https://ml-jku.github.io/UPT

MIM-Refiner: A Contrastive Learning Boost from Intermediate Pre-Trained Representations

We introduce MIM (Masked Image Modeling)-Refiner, a contrastive learning boost for pre-trained MIM models. The motivation behind MIM-Refiner is rooted in the insight that optimal representations within MIM models generally reside in intermediate layers. Accordingly, MIM-Refiner leverages multiple contrastive heads that are connected to diverse intermediate layers. In each head, a modified nearest neighbor objective helps to construct respective semantic clusters. The refinement process is short but effective. Within a few epochs, we refine the features of MIM models from subpar to state-of-the-art, off-the-shelf features. Refining a ViT-H, pre-trained with data2vec 2.0 on ImageNet-1K, achieves new state-of-the-art results in linear probing (84.7%) and low-shot classification among models that are pre-trained on ImageNet-1K. In ImageNet-1K 1-shot classification, MIM-Refiner sets a new state-of-the-art of 64.2%, outperforming larger models that were trained on up to 2000x more data such as DINOv2-g, OpenCLIP-G and MAWS-6.5B. Project page: https://ml-jku.github.io/MIM-Refiner

Neural SPH: Improved Neural Modeling of Lagrangian Fluid Dynamics

Smoothed particle hydrodynamics (SPH) is omnipresent in modern engineering and scientific disciplines. SPH is a class of Lagrangian schemes that discretize fluid dynamics via finite material points that are tracked through the evolving velocity field. Due to the particle-like nature of the simulation, graph neural networks (GNNs) have emerged as appealing and successful surrogates. However, the practical utility of such GNN-based simulators relies on their ability to faithfully model physics, providing accurate and stable predictions over long time horizons - which is a notoriously hard problem. In this work, we identify particle clustering originating from tensile instabilities as one of the primary pitfalls. Based on these insights, we enhance both training and rollout inference of state-of-the-art GNN-based simulators with varying components from standard SPH solvers, including pressure, viscous, and external force components. All neural SPH-enhanced simulators achieve better performance, often by orders of magnitude, than the baseline GNNs, allowing for significantly longer rollouts and significantly better physics modeling. Code available under (https://github.com/tumaer/neuralsph).

Lie Point Symmetry and Physics Informed Networks

Symmetries have been leveraged to improve the generalization of neural networks through different mechanisms from data augmentation to equivariant architectures. However, despite their potential, their integration into neural solvers for partial differential equations (PDEs) remains largely unexplored. We explore the integration of PDE symmetries, known as Lie point symmetries, in a major family of neural solvers known as physics-informed neural networks (PINNs). We propose a loss function that informs the network about Lie point symmetries in the same way that PINN models try to enforce the underlying PDE through a loss function. Intuitively, our symmetry loss ensures that the infinitesimal generators of the Lie group conserve the PDE solutions. Effectively, this means that once the network learns a solution, it also learns the neighbouring solutions generated by Lie point symmetries. Empirical evaluations indicate that the inductive bias introduced by the Lie point symmetries of the PDEs greatly boosts the sample efficiency of PINNs.

PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solvers

Time-dependent partial differential equations (PDEs) are ubiquitous in science and engineering. Recently, mostly due to the high computational cost of traditional solution techniques, deep neural network based surrogates have gained increased interest. The practical utility of such neural PDE solvers relies on their ability to provide accurate, stable predictions over long time horizons, which is a notoriously hard problem. In this work, we present a large-scale analysis of common temporal rollout strategies, identifying the neglect of non-dominant spatial frequency information, often associated with high frequencies in PDE solutions, as the primary pitfall limiting stable, accurate rollout performance. Based on these insights, we draw inspiration from recent advances in diffusion models to introduce PDE-Refiner; a novel model class that enables more accurate modeling of all frequency components via a multistep refinement process. We validate PDE-Refiner on challenging benchmarks of complex fluid dynamics, demonstrating stable and accurate rollouts that consistently outperform state-of-the-art models, including neural, numerical, and hybrid neural-numerical architectures. We further demonstrate that PDE-Refiner greatly enhances data efficiency, since the denoising objective implicitly induces a novel form of spectral data augmentation. Finally, PDE-Refiner's connection to diffusion models enables an accurate and efficient assessment of the model's predictive uncertainty, allowing us to estimate when the surrogate becomes inaccurate.