Alejandro F. Queiruga

Alejandro Francisco Queiruga

Scaling laws can be understood from ground-up numerical analysis, where traditional function approximation theory can explain shifts in model architecture choices. GLU variants now dominate frontier LLMs and similar outer-product architectures are prevalent in ranking models. The success of these architectures has mostly been left as an empirical discovery. In this paper, we apply the tools of numerical analysis to expose a key factor: these models have an $x^2$ which enables \emph{asymptotically} faster scaling than MLPs. GLUs have piecewise quadratic functional forms that are sufficient to exhibit quadratic order of approximation. Our key contribution is to demonstrate that the $L(P)$ scaling slope is $L(P)\propto P^{-3}$ for GLUs but only $L(P)=P^{-2}$ for MLPs on function reconstruction problems. We provide a parameter construction and empirical verification of these slopes for 1D function approximation. From the first principles we discover, we make one stride and propose the ``Gated Quadratic Unit'' which has an even steeper $L(P)$ slope than the GLU and MLP. This opens the possibility of architecture design from first principles numerical theory to unlock superior scaling in large models. Replication code is available at https://github.com/afqueiruga/divine_scaling.

Alejandro Francisco Queiruga, Theo Gutman-Solo, Shuai Jiang

While there are many applications of ML to scientific problems that look promising, visuals can be deceiving. For scientific applications, actual quantitative accuracy is crucial. This work applies the rigor of numerical analysis for differential equations to machine learning by specifically quantifying the accuracy of applying different ML techniques to the elementary 1D Poisson differential equation. Beyond the quantity and discretization of data, we identify that the function space of the data is critical to the generalization of the model. We prove generalization bounds and convergence rates under finite data discretizations and restricted training data subspaces by analyzing the training dynamics and deriving optimal parameters for both a white-box differential equation discovery method and a black-box linear model. The analytically derived generalization bounds are replicated empirically. Similar lack of generalization is empirically demonstrated for deep linear models, shallow neural networks, and physics-specific DeepONets and Neural Operators. We theoretically and empirically demonstrate that generalization to the true physical equation is not guaranteed in each explored case. Surprisingly, we find that different classes of models can exhibit opposing generalization behaviors. Based on our theoretical analysis, we also demonstrate a new mechanistic interpretability lens on scientific models whereby Green's function representations can be extracted from the weights of black-box models. Our results inform a new cross-validation technique for measuring generalization in physical systems. We propose applying it to the Poisson equation as an evaluation benchmark of future methods.

Aditi S. Krishnapriyan, Alejandro F. Queiruga, N. Benjamin Erichson et al.

Dynamical systems that evolve continuously over time are ubiquitous throughout science and engineering. Machine learning (ML) provides data-driven approaches to model and predict the dynamics of such systems. A core issue with this approach is that ML models are typically trained on discrete data, using ML methodologies that are not aware of underlying continuity properties. This results in models that often do not capture any underlying continuous dynamics -- either of the system of interest, or indeed of any related system. To address this challenge, we develop a convergence test based on numerical analysis theory. Our test verifies whether a model has learned a function that accurately approximates an underlying continuous dynamics. Models that fail this test fail to capture relevant dynamics, rendering them of limited utility for many scientific prediction tasks; while models that pass this test enable both better interpolation and better extrapolation in multiple ways. Our results illustrate how principled numerical analysis methods can be coupled with existing ML training/testing methodologies to validate models for science and engineering applications.

Alejandro Queiruga, N. Benjamin Erichson, Liam Hodgkinson et al.

The recently-introduced class of ordinary differential equation networks (ODE-Nets) establishes a fruitful connection between deep learning and dynamical systems. In this work, we reconsider formulations of the weights as continuous-in-depth functions using linear combinations of basis functions which enables us to leverage parameter transformations such as function projections. In turn, this view allows us to formulate a novel stateful ODE-Block that handles stateful layers. The benefits of this new ODE-Block are twofold: first, it enables incorporating meaningful continuous-in-depth batch normalization layers to achieve state-of-the-art performance; second, it enables compressing the weights through a change of basis, without retraining, while maintaining near state-of-the-art performance and reducing both inference time and memory footprint. Performance is demonstrated by applying our stateful ODE-Block to (a) image classification tasks using convolutional units and (b) sentence-tagging tasks using transformer encoder units.

Alejandro F. Queiruga, N. Benjamin Erichson, Dane Taylor et al.

Recent work has attempted to interpret residual networks (ResNets) as one step of a forward Euler discretization of an ordinary differential equation, focusing mainly on syntactic algebraic similarities between the two systems. Discrete dynamical integrators of continuous dynamical systems, however, have a much richer structure. We first show that ResNets fail to be meaningful dynamical integrators in this richer sense. We then demonstrate that neural network models can learn to represent continuous dynamical systems, with this richer structure and properties, by embedding them into higher-order numerical integration schemes, such as the Runge Kutta schemes. Based on these insights, we introduce ContinuousNet as a continuous-in-depth generalization of ResNet architectures. ContinuousNets exhibit an invariance to the particular computational graph manifestation. That is, the continuous-in-depth model can be evaluated with different discrete time step sizes, which changes the number of layers, and different numerical integration schemes, which changes the graph connectivity. We show that this can be used to develop an incremental-in-depth training scheme that improves model quality, while significantly decreasing training time. We also show that, once trained, the number of units in the computational graph can even be decreased, for faster inference with little-to-no accuracy drop.

N. Benjamin Erichson, Omri Azencot, Alejandro Queiruga et al.

Viewing recurrent neural networks (RNNs) as continuous-time dynamical systems, we propose a recurrent unit that describes the hidden state's evolution with two parts: a well-understood linear component plus a Lipschitz nonlinearity. This particular functional form facilitates stability analysis of the long-term behavior of the recurrent unit using tools from nonlinear systems theory. In turn, this enables architectural design decisions before experimentation. Sufficient conditions for global stability of the recurrent unit are obtained, motivating a novel scheme for constructing hidden-to-hidden matrices. Our experiments demonstrate that the Lipschitz RNN can outperform existing recurrent units on a range of benchmark tasks, including computer vision, language modeling and speech prediction tasks. Finally, through Hessian-based analysis we demonstrate that our Lipschitz recurrent unit is more robust with respect to input and parameter perturbations as compared to other continuous-time RNNs.