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Exact Representation of Piecewise Polynomial Finite Element Functions by Two-Hidden-Layer Multilayer Perceptrons

Jun Hu, Pengzhan Jin, Shengyang Xu

arXiv:2609.39856v1math.NAcs.NA

TL;DR

Finite element functions have explicit local polynomial structure, but exact links between these functions and neural-network representations are needed to connect their approximation properties. This paper constructs exact two-hidden-layer MLP representations across dimensions and degrees, with linear sparse storage and parameters computed without training. The constructions represent continuous functions pointwise and discontinuous functions almost everywhere, and two hidden layers are necessary in the worst case for the stated activation model.

  • Problem

    Exact neural representations of finite element functions connect neural-network and finite element approximation frameworks, whose relation matters for transferring finite element error estimates to network classes.

    Source: §1 · p. 1

  • Method

    The paper constructs two-hidden-layer MLPs using mesh geometry for localization and degree-dependent activations for exact polynomial reproduction, with parameters computed from mesh and finite element data.

    Source: §1 · p. 2, §5 · p. 21

  • Results

    For fixed dimension, degree, and output dimension, sparse storage grows linearly with element count; continuous functions admit pointwise representations, and two layers are necessary in the worst case for the stated activation model.

    Source: §1 · p. 3, §5 · p. 22

  • Takeaways & Limitations

    Exact realization transfers finite element approximation estimates to DG network realizations and to continuous network realizations in the stated mesh and Sobolev settings.

    Source: §5 · pp. 21–22

    The explicit coefficient maps may be ill-conditioned at higher polynomial degree.

    Source: §5 · p. 22

Abstract

from arXiv · show

We construct exact representations of finite element functions of arbitrary polynomial degree on simplicial meshes in any spatial dimension by two-hidden-layer multilayer perceptrons (MLPs). The first hidden layer uses $\mathrm{ReLU}^0+\mathrm{ReLU}$, and the second uses the degree-dependent activation $\mathrm{ReLU}^k$. The neural networks can realize both zero-skeleton and prescribed-skeleton representatives of discontinuous finite element functions, where the former vanish on the mesh skeleton and the latter admit independently prescribed polynomial data on each relatively open lower-dimensional mesh subsimplex; in particular, continuous finite element functions can be represented pointwise on the closed domain. In the continuous case, a half-open decomposition reduces the second-hidden-layer width by assigning each subsimplex to a single incident element. The number of nonzero parameters grows linearly with the number of elements for fixed spatial dimension, polynomial degree, and output dimension, and it is of the same order as the number of global degrees of freedom of the finite element space in the continuous case, provided that the mesh is shape-regular. All parameters in the neural network can be explicitly computed via the information of the mesh and finite element function without training.

1 Introduction

The paper gives exact two-hidden-layer MLP representations of finite element functions across dimensions and polynomial degrees, joining neural-network and finite element approximation frameworks. Its constructions combine prescribed pointwise behavior, linear sparse storage, and explicit parameters computed without training.

Uniform two-hidden-layer representation

  • The construction uses ReLU0 + ReLU in the first hidden layer and degree-dependent ReLUk in the second to represent functions on arbitrary conforming simplicial meshes.The first layer encodes mesh geometry, the second combines localization with polynomial reproduction, and the linear output layer assembles the function.Source: §1 · p. 2

Skeleton representations

  • Discontinuous functions are represented almost everywhere, while continuous functions are represented pointwise on the closed domain, with independently prescribed polynomial data also allowed on mesh subsimplices.A half-open decomposition reduces the continuous construction to one expansion per element when available.Source: §1 · pp. 2–3

Sparse storage

  • For fixed dimension, degree, and output dimension, sparse storage grows linearly with element count and, for scalar continuous functions on shape-regular meshes, is comparable to finite element space dimension.The half-open decomposition gives continuous functions the same second-layer width as the zero-skeleton discontinuous construction.Source: §1 · p. 2

Explicit parameter computation

  • All network parameters are computed explicitly from mesh geometry and finite element data, without training or nonlinear parameter fitting.Hidden weights and biases are independent of the target function; local output coefficients can be recovered from Lagrange nodal values using a reusable dimension- and degree-dependent matrix.Source: §1 · pp. 2–3

Approximation estimates

  • Exact realization transfers broken Sobolev approximation estimates to DG networks and global Sobolev estimates to continuous network approximants in the stated settings.The paper gives broken W^s,p estimates on shape-regular meshes and global W^s,p estimates for s = 0, 1 on uniform Kuhn meshes.Source: §1 · p. 3

Depth necessity

  • Two hidden layers are necessary in the worst case for the stated activation model because a one-hidden-layer network cannot match mixed differentiable and nondifferentiable subsets on one hyperplane.The counterexample is a continuous piecewise affine finite element function with both types of subsets having positive surface measure.Source: §1 · p. 3

2 Preliminaries

The preliminaries fix the mesh, finite element spaces, network architecture, and polynomial basis used in the constructions. They also specify how pointwise mesh-skeleton data and local polynomial coefficients are handled.

2.1 Simplicial meshes

  • The notation sets d as spatial dimension and k as polynomial degree, and the mesh is a conforming simplicial partition whose subsimplices partition its skeleton.The count of distinct facet-supporting hyperplanes identifies coplanar facets only once.Source: §2 · p. 3, §2.1 · p. 4

2.2 Finite element spaces

  • Finite element functions take values in a finite-dimensional space E, while discontinuous equivalence classes do not prescribe skeleton values and the paper distinguishes pointwise representatives.No compatibility is imposed between polynomial data on distinct relatively open subsimplices; continuity determines skeleton values from element polynomials.Source: §2.2 · p. 5

2.3 Neural network model

  • The two-hidden-layer network combines ϱ0 and ϱ1 in the first layer with ϱk in the second, and its pointwise outputs realize the specified function class.The paired first-layer parameterization uses two activation channels per affine preactivation, so the actual widths are (2n1, n2).Source: §2.3 · p. 6

2.4 Polynomials on a simplex

  • Affine powers of barycentric-coordinate expressions form a basis for degree-k polynomials on a simplex, enabling elementwise polynomial reproduction.Local coefficients can be recovered from nodal values through a reference matrix independent of element geometry, and its factorization can be reused across elements.Source: §2.4 · pp. 7–8

3 Pointwise realization of discontinuous finite element functions

The section constructs exact two-hidden-layer representations of discontinuous finite element functions, both with zero skeleton values and with independently prescribed polynomial data on mesh subsimplices. It extends the construction to vector- and tensor-valued spaces.

3.1 Scalar zero-skeleton representation

  • The scalar construction realizes zero-skeleton representatives exactly, with second-layer width MNT and at most 2HT distinct first-layer affine preactivations.These representatives equal the element polynomial in each open element and vanish on the mesh skeleton.Source: §3.1 · pp. 9–10

3.2 Vector- and tensor-valued finite elements

  • Shared hidden features represent vector- and tensor-valued functions componentwise, including constrained spaces when k bounds their local polynomial degree.For H(div) and H(curl), the zero-skeleton representative preserves the Sobolev class and traces, though its assigned skeleton values need not equal those traces.Source: §3.2 · pp. 10–11

3.3 Pointwise representation with prescribed skeleton values

  • Independent polynomial data on each relatively open subsimplex yield a pointwise-exact representation across the domain using second-layer width MNS.For fixed dimension and degree, hidden parameters depend on the mesh and the chosen incident element for each subsimplex; only output coefficients vary with the function.Source: §3.3 · p. 14

4 Pointwise realization of continuous finite element functions

This section presents neural-network realizations of continuous finite element functions, including pointwise representation of mesh-skeleton values and approximation in Lp away from shrinking skeleton neighborhoods. It also discusses width reduction through half-open decompositions, sparse-storage scaling, and global approximation bounds on uniform meshes.

4.1 Continuous representations

  • A half-open decomposition assigns each mesh subsimplex to one incident element, reducing second-layer width from MNS to MNT without separate lower-dimensional expansions.Incident element polynomials agree on shared faces, so each assigned element polynomial supplies the boundary-face values as well as its interior values.Source: §4.1 · p. 15
  • Two-hidden-layer networks realize continuous finite element functions pointwise on the closed domain, including all interior mesh-skeleton and exterior-boundary values.The general construction uses second-layer width MNS; the half-open construction uses MNT.Source: §4.1 · p. 15, §4.2 · p. 18
  • The half-open width reduction requires a decomposition, which the star condition guarantees but is not necessary for; such decompositions need not exist on arbitrary meshes.Every bounded convex polyhedral domain satisfies the star condition; in two dimensions, disk- or annulus-type polygonal domains also admit the decomposition, including nonconvex examples.Source: §4.1 · pp. 16–18
  • If boundary values are not prescribed, any conforming mesh admits pointwise representation throughout Ω, including interior skeleton values, without a star condition.The construction makes no assertion about ∂Ω values.Source: §4.1 · p. 18

4.2 Width and sparse storage

  • For fixed dimension and degree, sparse parameter storage is O_d,k(NT), and under shape regularity it is comparable to the continuous finite element space dimension.The retained-storage comparison is two-sided; individual parameter values may vanish for particular target functions.Source: §4.2 · pp. 19–20

4.3 Approximation on a uniform mesh

  • For h = 1/N on Ω = (0, 1)^d, every v ∈ W^{r,p}(Ω; E) has a network N with ||v − N||_{W^{s,p}(Ω;E)} ≤ C h^{r−s} |v|_{W^{r,p}(Ω;E)} for s = 0, 1.The result assumes k ≥ 1, integer 1 ≤ r ≤ k + 1, and 1 ≤ p ≤ ∞; C depends only on d, k, r, p, and m.Source: §4.3 · pp. 20–21
  • For each fixed degree-k finite element function, these networks agree with it outside arbitrarily thin mesh-skeleton neighborhoods, remain uniformly bounded as the neighborhoods shrink, and converge to it in Lp.The construction uses β and ReLU^k in the second hidden layer.Source: §4.3 · p. 21
  • The localization construction makes hidden-layer widths independent of neighborhood thickness, while weights may grow and the resulting networks yield the corresponding Lp approximation rates.Source: §4.3 · p. 21

5 Conclusion

The conclusion gives a constructive correspondence between piecewise polynomial finite element functions and two-hidden-layer MLPs, with exact representations on conforming simplicial meshes. It highlights linear sparse storage, transferred approximation estimates, and open issues in coefficient computation and representation design.

5 Conclusion

  • The construction uses geometric tests to set skeleton values, yielding almost-everywhere DG representations and pointwise continuous representations on the closed domain.It applies in arbitrary dimension and polynomial degree, with common hidden features for scalar, vector, and tensor outputs.Source: §5 · p. 21
  • For fixed dimension, degree, and output dimension, sparse storage grows linearly with element count; under shape regularity, scalar continuous storage is comparable to finite-element dimension.A half-open decomposition reduces the continuous construction to one expansion per element.Source: §5 · p. 22
  • Exact realization transfers broken W^s,p approximation estimates to DG networks and global W^1,p estimates to continuous networks without additional representation error.Parameters come from geometric formulas and local linear systems, with reusable reference factorizations.Source: §5 · p. 22
  • The construction uses a discontinuous first-layer activation and degree-dependent second-layer activation; explicit coefficient maps may be ill-conditioned at higher degree.Further work includes stable coefficient computation, weight-growth control, and smaller pointwise representations on general meshes.Source: §5 · p. 22

A Local polynomial coefficients

The appendix computes local polynomial coefficients through a degree-dependent matrix system, whose structure enables reusable triangular-solve computations. It also warns that the matrix can become ill-conditioned at higher degrees.

A Local polynomial coefficients

  • For k = 0, the sole coefficient is the constant value of the polynomial.For k >= 1, the appendix instead sets up a coefficient computation based on the degree-k Lagrange lattice.Source: A Local polynomial coefficients · p. 22
  • The degree-k Lagrange nodes have barycentric coordinates bα/k, and the coefficient vector u satisfies p = Au.The matrix factorization A = V^T Dk V, with positive diagonal Dk, implies that A is symmetric positive definite.Source: A Local polynomial coefficients · p. 22
  • Newton polynomials give a unit lower triangular coefficient matrix L; the resulting triangular factors reduce coefficient computation to triangular solves and scaling.The matrices are independent of the element geometry and can be prepared once and reused across elements; the same factorization applies componentwise to vector- and tensor-valued polynomials.Source: A Local polynomial coefficients · pp. 22–23
  • For d = 2 and k = 6, A has dimension 28 and Euclidean condition number approximately 4.68 × 10^9, which may cause numerical stability issues.The warning applies even when an explicit inverse is avoided.Source: A Local polynomial coefficients · p. 23
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