Clifford Neural Layers for PDE Modeling

Partial differential equations (PDEs) see widespread use in sciences and engineering to describe simulation of physical processes as scalar and vector fields interacting and coevolving over time. Due to the computationally expensive nature of their standard solution methods, neural PDE surrogates have become an active research topic to accelerate these simulations. However, current methods do not explicitly take into account the relationship between different fields and their internal components, which are often correlated. Viewing the time evolution of such correlated fields through the lens of multivector fields allows us to overcome these limitations. Multivector fields consist of scalar, vector, as well as higher-order components, such as bivectors and trivectors. Their algebraic properties, such as multiplication, addition and other arithmetic operations can be described by Clifford algebras. To our knowledge, this paper presents the first usage of such multivector representations together with Clifford convolutions and Clifford Fourier transforms in the context of deep learning. The resulting Clifford neural layers are universally applicable and will find direct use in the areas of fluid dynamics, weather forecasting, and the modeling of physical systems in general. We empirically evaluate the benefit of Clifford neural layers by replacing convolution and Fourier operations in common neural PDE surrogates by their Clifford counterparts on two-dimensional Navier-Stokes and weather modeling tasks, as well as three-dimensional Maxwell equations. Clifford neural layers consistently improve generalization capabilities of the tested neural PDE surrogates.

Bayesian Optimization for Macro Placement

Macro placement is the problem of placing memory blocks on a chip canvas. It can be formulated as a combinatorial optimization problem over sequence pairs, a representation which describes the relative positions of macros. Solving this problem is particularly challenging since the objective function is expensive to evaluate. In this paper, we develop a novel approach to macro placement using Bayesian optimization (BO) over sequence pairs. BO is a machine learning technique that uses a probabilistic surrogate model and an acquisition function that balances exploration and exploitation to efficiently optimize a black-box objective function. BO is more sample-efficient than reinforcement learning and therefore can be used with more realistic objectives. Additionally, the ability to learn from data and adapt the algorithm to the objective function makes BO an appealing alternative to other black-box optimization methods such as simulated annealing, which relies on problem-dependent heuristics and parameter-tuning. We benchmark our algorithm on the fixed-outline macro placement problem with the half-perimeter wire length objective and demonstrate competitive performance.

Path Integral Stochastic Optimal Control for Sampling Transition Paths

We consider the problem of Sampling Transition Paths. Given two metastable conformational states of a molecular system, eg. a folded and unfolded protein, we aim to sample the most likely transition path between the two states. Sampling such a transition path is computationally expensive due to the existence of high free energy barriers between the two states. To circumvent this, previous work has focused on simplifying the trajectories to occur along specific molecular descriptors called Collective Variables (CVs). However, finding CVs is not trivial and requires chemical intuition. For larger molecules, where intuition is not sufficient, using these CV-based methods biases the transition along possibly irrelevant dimensions. Instead, this work proposes a method for sampling transition paths that consider the entire geometry of the molecules. To achieve this, we first relate the problem to recent work on the Schrodinger bridge problem and stochastic optimal control. Using this relation, we construct a method that takes into account important characteristics of molecular systems such as second-order dynamics and invariance to rotations and translations. We demonstrate our method on the commonly studied Alanine Dipeptide, but also consider larger proteins such as Polyproline and Chignolin.

Complex-Valued Autoencoders for Object Discovery

Object-centric representations form the basis of human perception and enable us to reason about the world and to systematically generalize to new settings. Currently, most machine learning work on unsupervised object discovery focuses on slot-based approaches, which explicitly separate the latent representations of individual objects. While the result is easily interpretable, it usually requires the design of involved architectures. In contrast to this, we propose a distributed approach to object-centric representations: the Complex AutoEncoder. Following a coding scheme theorized to underlie object representations in biological neurons, its complex-valued activations represent two messages: their magnitudes express the presence of a feature, while the relative phase differences between neurons express which features should be bound together to create joint object representations. We show that this simple and efficient approach achieves better reconstruction performance than an equivalent real-valued autoencoder on simple multi-object datasets. Additionally, we show that it achieves competitive unsupervised object discovery performance to a SlotAttention model on two datasets, and manages to disentangle objects in a third dataset where SlotAttention fails - all while being 7-70 times faster to train.

Equivariant Diffusion for Molecule Generation in 3D

This work introduces a diffusion model for molecule generation in 3D that is equivariant to Euclidean transformations. Our E(3) Equivariant Diffusion Model (EDM) learns to denoise a diffusion process with an equivariant network that jointly operates on both continuous (atom coordinates) and categorical features (atom types). In addition, we provide a probabilistic analysis which admits likelihood computation of molecules using our model. Experimentally, the proposed method significantly outperforms previous 3D molecular generative methods regarding the quality of generated samples and efficiency at training time.

Adversarial Defense via Image Denoising with Chaotic Encryption

In the literature on adversarial examples, white box and black box attacks have received the most attention. The adversary is assumed to have either full (white) or no (black) access to the defender's model. In this work, we focus on the equally practical gray box setting, assuming an attacker has partial information. We propose a novel defense that assumes everything but a private key will be made available to the attacker. Our framework uses an image denoising procedure coupled with encryption via a discretized Baker map. Extensive testing against adversarial images (e.g. FGSM, PGD) crafted using various gradients shows that our defense achieves significantly better results on CIFAR-10 and CIFAR-100 than the state-of-the-art gray box defenses in both natural and adversarial accuracy.

Defending Variational Autoencoders from Adversarial Attacks with MCMC

Variational autoencoders (VAEs) are deep generative models used in various domains. VAEs can generate complex objects and provide meaningful latent representations, which can be further used in downstream tasks such as classification. As previous work has shown, one can easily fool VAEs to produce unexpected latent representations and reconstructions for a visually slightly modified input. Here, we examine several objective functions for adversarial attacks construction, suggest metrics assess the model robustness, and propose a solution to alleviate the effect of an attack. Our method utilizes the Markov Chain Monte Carlo (MCMC) technique in the inference step and is motivated by our theoretical analysis. Thus, we do not incorporate any additional costs during training or we do not decrease the performance on non-attacked inputs. We validate our approach on a variety of datasets (MNIST, Fashion MNIST, Color MNIST, CelebA) and VAE configurations ($\beta$-VAE, NVAE, TC-VAE) and show that it consistently improves the model robustness to adversarial attacks.

Neural RF SLAM for unsupervised positioning and mapping with channel state information

We present a neural network architecture for jointly learning user locations and environment mapping up to isometry, in an unsupervised way, from channel state information (CSI) values with no location information. The model is based on an encoder-decoder architecture. The encoder network maps CSI values to the user location. The decoder network models the physics of propagation by parametrizing the environment using virtual anchors. It aims at reconstructing, from the encoder output and virtual anchor location, the set of time of flights (ToFs) that are extracted from CSI using super-resolution methods. The neural network task is set prediction and is accordingly trained end-to-end. The proposed model learns an interpretable latent, i.e., user location, by just enforcing a physics-based decoder. It is shown that the proposed model achieves sub-meter accuracy on synthetic ray tracing based datasets with single anchor SISO setup while recovering the environment map up to 4cm median error in a 2D environment and 15cm in a 3D environment

Lie Point Symmetry Data Augmentation for Neural PDE Solvers

Neural networks are increasingly being used to solve partial differential equations (PDEs), replacing slower numerical solvers. However, a critical issue is that neural PDE solvers require high-quality ground truth data, which usually must come from the very solvers they are designed to replace. Thus, we are presented with a proverbial chicken-and-egg problem. In this paper, we present a method, which can partially alleviate this problem, by improving neural PDE solver sample complexity -- Lie point symmetry data augmentation (LPSDA). In the context of PDEs, it turns out that we are able to quantitatively derive an exhaustive list of data transformations, based on the Lie point symmetry group of the PDEs in question, something not possible in other application areas. We present this framework and demonstrate how it can easily be deployed to improve neural PDE solver sample complexity by an order of magnitude.

Message Passing Neural PDE Solvers

The numerical solution of partial differential equations (PDEs) is difficult, having led to a century of research so far. Recently, there have been pushes to build neural--numerical hybrid solvers, which piggy-backs the modern trend towards fully end-to-end learned systems. Most works so far can only generalize over a subset of properties to which a generic solver would be faced, including: resolution, topology, geometry, boundary conditions, domain discretization regularity, dimensionality, etc. In this work, we build a solver, satisfying these properties, where all the components are based on neural message passing, replacing all heuristically designed components in the computation graph with backprop-optimized neural function approximators. We show that neural message passing solvers representationally contain some classical methods, such as finite differences, finite volumes, and WENO schemes. In order to encourage stability in training autoregressive models, we put forward a method that is based on the principle of zero-stability, posing stability as a domain adaptation problem. We validate our method on various fluid-like flow problems, demonstrating fast, stable, and accurate performance across different domain topologies, discretization, etc. in 1D and 2D. Our model outperforms state-of-the-art numerical solvers in the low resolution regime in terms of speed and accuracy.