PDE constraints on smooth hierarchical functions computed by neural networks

• URL: http://arxiv.org/abs/2005.08859v2
• Date: Fri, 13 Aug 2021 17:58:10 GMT
• Title: PDE constraints on smooth hierarchical functions computed by neural networks
• Authors: Khashayar Filom, Konrad Paul Kording, Roozbeh Farhoodi
• Abstract summary: An important problem in the theory of deep neural networks is expressivity. We study real infinitely differentiable (smooth) hierarchical functions implemented by feedforward neural networks. We conjecture that such PDE constraints, once accompanied by appropriate non-singularity conditions, guarantee that the smooth function under consideration can be represented by the network.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Neural networks are versatile tools for computation, having the ability to approximate a broad range of functions. An important problem in the theory of deep neural networks is expressivity; that is, we want to understand the functions that are computable by a given network. We study real infinitely differentiable (smooth) hierarchical functions implemented by feedforward neural networks via composing simpler functions in two cases: 1) each constituent function of the composition has fewer inputs than the resulting function; 2) constituent functions are in the more specific yet prevalent form of a non-linear univariate function (e.g. tanh) applied to a linear multivariate function. We establish that in each of these regimes there exist non-trivial algebraic partial differential equations (PDEs), which are satisfied by the computed functions. These PDEs are purely in terms of the partial derivatives and are dependent only on the topology of the network. For compositions of polynomial functions, the algebraic PDEs yield non-trivial equations (of degrees dependent only on the architecture) in the ambient polynomial space that are satisfied on the associated functional varieties. Conversely, we conjecture that such PDE constraints, once accompanied by appropriate non-singularity conditions and perhaps certain inequalities involving partial derivatives, guarantee that the smooth function under consideration can be represented by the network. The conjecture is verified in numerous examples including the case of tree architectures which are of neuroscientific interest. Our approach is a step toward formulating an algebraic description of functional spaces associated with specific neural networks, and may provide new, useful tools for constructing neural networks.

Related papers

• Polynomial-Augmented Neural Networks (PANNs) with Weak Orthogonality Constraints for Enhanced Function and PDE Approximation [22.689531776611084]
  We present-augmented neural networks (PANNs) We present a novel machine learning architecture that combines deep neural networks (DNNs) with an approximant.
  arXiv  Detail & Related papers  (2024-06-04T14:06:15Z)
• Structure of universal formulas [13.794391803767617]
  We introduce a hierarchy of classes connecting the global approximability property to the weaker property of infinite VC dimension. We show that fixed-size neural networks with not more than one layer of neurons having activations cannot approximate functions on arbitrary finite sets. We give examples of functional families, including two-hidden-layer neural networks, that approximate functions on arbitrary finite sets, but fail to do that on the whole domain of definition.
  arXiv  Detail & Related papers  (2023-11-07T11:50:25Z)
• Data-driven discovery of Green's functions [0.0]
  This thesis introduces theoretical results and deep learning algorithms to learn Green's functions associated with linear partial differential equations. The construction connects the fields of PDE learning and numerical linear algebra. Rational neural networks (NNs) are introduced and consist of neural networks with trainable rational activation functions.
  arXiv  Detail & Related papers  (2022-10-28T09:41:50Z)
• Learning differentiable solvers for systems with hard constraints [48.54197776363251]
  We introduce a practical method to enforce partial differential equation (PDE) constraints for functions defined by neural networks (NNs) We develop a differentiable PDE-constrained layer that can be incorporated into any NN architecture. Our results show that incorporating hard constraints directly into the NN architecture achieves much lower test error when compared to training on an unconstrained objective.
  arXiv  Detail & Related papers  (2022-07-18T15:11:43Z)
• Benefits of Overparameterized Convolutional Residual Networks: Function Approximation under Smoothness Constraint [48.25573695787407]
  We prove that large ConvResNets can not only approximate a target function in terms of function value, but also exhibit sufficient first-order smoothness. Our theory partially justifies the benefits of using deep and wide networks in practice.
  arXiv  Detail & Related papers  (2022-06-09T15:35:22Z)
• Message Passing Neural PDE Solvers [60.77761603258397]
  We build a neural message passing solver, replacing allally designed components in the 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. We validate our method on various fluid-like flow problems, demonstrating fast, stable, and accurate performance across different domain topologies, equation parameters, discretizations, etc., in 1D and 2D.
  arXiv  Detail & Related papers  (2022-02-07T17:47:46Z)
• Neural Operator: Learning Maps Between Function Spaces [75.93843876663128]
  We propose a generalization of neural networks to learn operators, termed neural operators, that map between infinite dimensional function spaces. We prove a universal approximation theorem for our proposed neural operator, showing that it can approximate any given nonlinear continuous operator. An important application for neural operators is learning surrogate maps for the solution operators of partial differential equations.
  arXiv  Detail & Related papers  (2021-08-19T03:56:49Z)
• Neural Network Approximation of Refinable Functions [8.323468006516018]
  We show that refinable functions are approximated by the outputs of deep ReLU networks with a fixed width and increasing depth with accuracy exponential. Our results apply to functions used in the standard construction of wavelets as well as to functions constructed via subdivision algorithms in Computer Aided Geometric Design.
  arXiv  Detail & Related papers  (2021-07-28T06:45:36Z)
• Neural Network Approximations of Compositional Functions With Applications to Dynamical Systems [3.660098145214465]
  We develop an approximation theory for compositional functions and their neural network approximations. We identify a set of key features of compositional functions and the relationship between the features and the complexity of neural networks. In addition to function approximations, we prove several formulae of error upper bounds for neural networks.
  arXiv  Detail & Related papers  (2020-12-03T04:40:25Z)
• Multipole Graph Neural Operator for Parametric Partial Differential Equations [57.90284928158383]
  One of the main challenges in using deep learning-based methods for simulating physical systems is formulating physics-based data. We propose a novel multi-level graph neural network framework that captures interaction at all ranges with only linear complexity. Experiments confirm our multi-graph network learns discretization-invariant solution operators to PDEs and can be evaluated in linear time.
  arXiv  Detail & Related papers  (2020-06-16T21:56:22Z)
• Neural Operator: Graph Kernel Network for Partial Differential Equations [57.90284928158383]
  This work is to generalize neural networks so that they can learn mappings between infinite-dimensional spaces (operators) We formulate approximation of the infinite-dimensional mapping by composing nonlinear activation functions and a class of integral operators. Experiments confirm that the proposed graph kernel network does have the desired properties and show competitive performance compared to the state of the art solvers.
  arXiv  Detail & Related papers  (2020-03-07T01:56:20Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.