PaperlayerView exact revision on arXiv

Source-linked AI summary

FOSLS-deRhaNN: native de Rham neural classes for H(div) and H(curl) with applications to first-order system least-squares neural network methods for partial differential equations

Shun Zhang

arXiv:2610.08016v1math.NAcs.LGcs.NA

TL;DR

For discontinuous coefficients, second-order strong-form residuals can require second derivatives and pointwise coefficient differentiation, while also needing explicit interface treatment. The paper constructs conforming neural classes in H(div) and H(curl) from fixed de Rham operators, with kinked potentials carrying admissible jumps. Its FOSLS functional is equivalent to error in natural norms under the stated weak-problem assumptions, including for H−1 right-hand sides and H1/2 Dirichlet data.

  • Problem

    For discontinuous coefficients, second-order strong-form residuals require second derivatives and pointwise coefficient differentiation, while also needing explicit interface treatment.

    Source: §1 · p. 2

  • Method

    The authors build conforming H(div) and H(curl) neural classes by applying fixed de Rham operators to scalar or componentwise networks, with potential terms carrying jumps.

    Source: §1.2 · p. 3, §1.3 · p. 5, §11 · p. 40

  • Results

    The classes are conforming for every parameter value, and kinked potentials represent admissible interface jumps at finite width.

    Source: §1.3 · p. 5, §11 · p. 40

  • Takeaways & Limitations

    For uniquely solvable general elliptic weak problems with bounded coefficients and uniformly positive definite A, the FOSLS functional is equivalent to natural-norm error, including for H−1 forcing and H1/2 Dirichlet data.

    Source: §1 · p. 2

    Conformity alone does not ensure vanishing best-approximation error; the union of the classes must be dense in the target space X.

    Source: §6.2 · p. 25

Abstract

from arXiv · show

We construct neural approximation classes native to the graph spaces H(div) and H(curl), in two and three dimensions and, for H(div), in any dimension. Every realization lies in the space for all parameter values, and with kinked potentials, such as ReLU networks, the admissible jumps appear at finite width. The classes are images of scalar and componentwise networks under fixed operators of the de Rham complex, and do not involve a mesh or finite element emulation. For H(div) in R^n two native classes are given on an equal footing, with a skew-symmetric potential $A$: $\mathrm{Div}\,A+R_nq+\mathbf{h}$, with the divergence $q$ as an explicit unknown, and $\mathrm{Div}\,A+\mathbf{z}$ with an $H^1$ field $\mathbf{z}$; for H(curl) the analogous classes are $\mathrm{grad}\,\phi+Sr+\mathbf{h}$ in two dimensions and $\mathrm{grad}\,\phi+\mathbf{z}$ in two and three dimensions. In all of them every interface jump of the field is carried by the potential term, $\mathrm{Div}\,A$ or $\mathrm{grad}\,\phi$, while the remaining part has no interface jump (it is an $H^1$ field in the regular-decomposition classes); the classes with $\mathbf{z}$ are the componentwise approach enriched by this term. Known or learned interface geometry enters the potential through factors with trainable amplitudes, and the remaining part if the divergence jumps. The classes lead to the FOSLS-deRhaNN method, first-order system least squares with de Rham neural networks, whose loss is the least-squares functional posed in the natural spaces of the weak formulation; for elliptic equations this includes $H^{-1}$ right-hand sides and $H^{1/2}$ Dirichlet data. Elliptic equations with discontinuous coefficients and curl-curl problems are treated as instances, with the functional equivalent to the error; linear transport with discontinuous solutions and conservation laws with shocks use the same flux classes.

1. Introduction

First-order formulations preserve the natural continuity of fluxes and gradients across discontinuous coefficients, but componentwise networks impose stronger continuity and cannot represent admissible jumps at finite size. The paper constructs de Rham neural classes that carry those jumps through potentials and combines them with first-order system least squares at the weak formulation’s natural regularity.

1. Introduction

  • First-order systems keep potential and flux as unknowns in their natural spaces, separating constitutive and conservation residuals and accommodating weak data that may not define a strong form.For elliptic problems, the right-hand side may lie in H^-1 and the Dirichlet data in H^1/2; the FOSLS functional is equivalent to error in natural norms under the stated solvability conditions.Source: §1 · pp. 1–2

1.1. Natural graph spaces and intrinsic continuity.

  • Componentwise continuous networks are conforming but overconstrain H(div) and H(curl), excluding fields with admissible jumps at finite network size.The de Rham classes supply those jumps through potential terms rather than approximating them only by narrowing continuous transition layers.Source: §1.1 · p. 2

1.2. de Rham representations.

  • H(div) fields admit two representations, Div A + R_nq + h with explicit divergence q, or Div A + z with an H^1 remainder; H(curl) uses corresponding gradient-potential representations.The classes are assembled from scalar and componentwise networks through fixed de Rham operators; conformity holds for every parameter value without a mesh or finite-element interpolation.Source: §1.2 · p. 3

1.3. Related work and outline.

  • The deRhaNN classes carry intrinsic graph-space conformity and finite-width jumps, extending compatible neural constructions whose structure is fixed by a mesh or neuron set.Combined with FOSLS, they yield a best-approximation error estimate; the paper treats elliptic and curl-curl problems, linear transport, and conservation laws with unknown shocks.Source: §1.3 · pp. 4–5

2. Native neural approximation in H(div) I: right inverse and stable de Rham representation

The section develops native H(div) neural classes assembled from de Rham potentials and divergence components, using a stable representation involving a skew-symmetric potential, a divergence right inverse, and a finite-dimensional cohomology component. It establishes conformity and density in H(div), describing scalar or vector curl potentials in two and three dimensions and relating finite realizations to Raviart–Thomas spaces.

2.1. Motivation: the de Rham sequence.

  • Every H(div) flux is represented by a divergence-free potential term, an independent divergence q lifted by R_n, and a finite-dimensional topological component h.The divergence determines q, while h accounts for topology; the potential represents the remaining divergence-free part.Source: §2.1 · p. 5

2.2. Skew-symmetric potentials.

  • Row-wise divergence of skew-symmetric potentials generalizes curl to R^n, is divergence-free, and has tangential interface jumps because its normal jump vanishes.In two and three dimensions, Div A reduces to the scalar and vector curl, respectively.Source: §2.2 · p. 6

2.3. A right inverse of the divergence.

  • A radial segment-integral right inverse satisfies div R_n q = q; under measure-zero interface and boundary crossings, it is continuous and carries no jumps.This applies, for example, to convex polygons or polyhedra with the center in the interior.Source: §2.3 · pp. 7–8

2.4. Cohomology part.

  • The finite-dimensional cohomology component encodes cavity fluxes, while the radial part has zero cavity flux and carries source balance through the outer boundary.The cohomology obstruction to a global skew potential is associated with cavities.Source: §2.4 · p. 8

2.5. Stable splitting and regular potentials.

  • A bounded direct splitting yields a norm-equivalent H(div) representation after quotienting out the potential kernel.Divergence-free fields with zero cavity flux admit potentials with one additional Sobolev derivative.Source: §2.5 · p. 10

2.6. Neural classes.

  • The neural class consists of Div A_m + R_n q_m + h, with scalar networks for the skew-potential entries and q_m, assembled using fixed operators.The H(div) field itself is not an unconstrained componentwise network.Source: §2.6 · p. 10
  • The class τ_m = Div A_m + R_n q_m + h lies in H(div; Ω) with div τ_m = q_m, and every τ has a corresponding potential representation.Here q = div τ and h = Π_Hn τ.Source: §2.6 · p. 10
  • The neural H(div) classes are dense: every τ in H(div; Ω) can be approximated with H(div) error tending to zero as network size m increases.Source: §2.6 · p. 10

2.7. Two and three dimensions.

  • In dimensions two and three, every H(div) field has a curl-potential representation with q = div τ and a harmonic component h; the potentials are unique up to their stated kernels and admit norm bounds.The representations are τ = curl ψ + R_2 q + h in two dimensions and τ = curl Ψ + R_3 q + h in three dimensions.Source: §2.7 · p. 11
  • On a 2D triangulation, divergence-free RT0 fields are exactly curls of continuous piecewise linear functions, which are ReLU networks; general RT0 fields are not asserted to have exact finite realization.For general RT0 fields, the representation reduces fitting to approximation of a scalar potential and the piecewise-constant divergence; in 3D, lowest-order Nédélec fields may be discontinuous.Source: §2.7 · p. 12

3. Native neural approximation in H(div) II: regular decomposition

A regular decomposition represents every H(div) field as a divergence-free potential term plus an H1 field, with controlled norms. The resulting neural class preserves H(div) conformity while adding admissible tangential jumps to componentwise networks.

3.1. Regular decomposition.

  • Every H(div) field admits a stable decomposition τ = Div A + z with A and z in H1; for zero-normal-trace fields, both components can satisfy homogeneous boundary conditions.The divergence is carried entirely by z, and harmonic fields are absorbed into it.Source: §3.1 · p. 12

3.2. The second neural class.

  • The second neural class uses τ_m = Div A_m + z_m, which lies in H(div) and approximates the decomposition through H1 approximation of both A and z.In two and three dimensions, the potential term is curl ψ_m or curl Ψ_m, respectively.Source: §3.2 · pp. 12–13

3.3. The two classes.

  • The two classes place divergence differently: the first makes q an independent L2 unknown, while the second uses div z and avoids line integrals and harmonic fields.In both, the potential carries interface jumps; the H1 remainder has no interface jump, and Div A enriches componentwise networks with admissible tangential jumps without changing divergence.Source: §3.3 · p. 13

4. Native neural approximation in H(div) III: finite-width jumps and interface-adapted potentials

This section explains how kinked de Rham potentials and interface factors in H(div) modules represent finite-width flux jumps, and how fixed, learned, or activation-defined interfaces encode their geometry. It also addresses potential regularity after extracting fixed-interface kinks and distinguishes flux jumps carried by the skew potential from divergence jumps handled by the remaining field or an analogous module.

4.1. Tangential jumps carried by kinks of the potential.

  • Kinks in the H1 potential carry admissible tangential flux jumps while preserving normal continuity; finite ReLU potentials yield piecewise-constant H(div) fields with jumps on activation faces.Smooth activations cannot represent such jumps exactly at finite width, and generic activation faces match curved interfaces only approximately.Source: §4.1 · pp. 13–14, §4 · p. 13

4.2. Fixed-interface potential modules.

  • A fixed-interface factor |dΓ|N places the kink on known geometry and realizes every Lipschitz tangential jump with an appropriate amplitude.For neural amplitudes, the jump is approximated as the network approximates the amplitude in H1.Source: §4.2 · p. 15

4.3. Regularity recovery and enriched approximation.

  • With a supported H2 amplitude matching the jump, extracting the fixed-interface kink leaves an H2 potential remainder; the jump must satisfy this amplitude hypothesis.The interface singularity is isolated in the explicit kink factor, leaving smooth-function approximation problems; the remaining part q or z is unaffected.Source: §4.3 · p. 16

4.4. Learned-interface potentials.

  • A learned ReLU interface preserves H(div) conformity for every parameter value, and at regular interface points its amplitude-gradient product generates tangential jumps.The results establish a local jump mechanism, not global interface recovery or an improved approximation rate for the learned-interface level.Source: §4.4 · pp. 16–17

4.5. Two and three dimensions.

  • In two dimensions, a scalar potential’s normal derivative kink gives the tangential curl jump; in three dimensions, the curl jump is tangential and determined by the potential kink.The fixed-interface and learned-interface modules yield the corresponding jump formulas in both dimensions.Source: §4.5 · pp. 17–18

4.6. A divergence that jumps.

  • For a jumping divergence, the q-based class permits q_m to jump while R_nq_m remains continuous, whereas the z-based class keeps z_m continuous and carries the divergence jump through a kink.The first class requires only q_m in L2 for conformity; the second uses an H1 field with no interface jump.Source: §4.6 · p. 18
  • For a prescribed Lipschitz divergence jump j on a C1,1 interface, z=z0+|dΓ|Z1 realizes it when 2Z1·grad dΓ=j; no z-module is needed if the divergence is continuous.For a learned interface, the corresponding module uses ReLU(gη)Z1.Source: §4.6 · p. 18

4.7. Three implementation levels.

  • With a known interface, A=A0+|dΓ|N produces the flux jump 2N grad dΓ; the geometry is fixed data and adds no trainable parameters.The product is differentiated piecewise, and quadrature is split along the known interface because the residual integrand may jump there.Source: §4.7 · pp. 18–19
  • With a learned interface, A=A0+ReLU(gη)N encodes geometry in gη and produces the flux jump N grad gη at regular interface points.Because the kink moves with η, integration must account for the moving zero set, by cutting quadrature cells or treating derivatives cellwise weakly.Source: §4.7 · p. 19
  • In the generic level, ReLU activation interfaces can create finite-width kinks, whereas smooth activations have no exact finite-width jump and only approximate one through a smooth transition layer.The geometry is implicit in the activation partition; a hidden preactivation zero set produces no output kink if its downstream coefficients vanish.Source: §4.7 · p. 19
  • If the target divergence is continuous across the interface, the remaining part is unchanged across all three levels; if it jumps, it is handled with the module described above.In two and three dimensions, ψ and Ψ replace A.Source: §4.7 · p. 19

5. Native neural approximation in H(curl)

This section constructs native H(curl) neural classes from scalar gradient potentials and jump-free remainders, with every realization in H(curl) for all parameter values. The structured classes preserve tangential continuity while allowing admissible normal jumps at finite width.

5.1. Motivation: the de Rham sequence.

  • In 2D, both curl-explicit and regular-decomposition classes are available; in 3D, only the regular decomposition is used because curl is not onto.In 2D the classes are grad ϕ + Sr + hcurl, with r = curl E, and grad ϕ + z, with z in H1; in 3D the class is grad ϕ + z.Source: §5.1 · p. 19, §5.2 · p. 20

5.2. Two dimensions.

  • In 2D, both neural classes remain in H(curl) for every parameter value and approximate fields through their scalar, curl, and regular-remainder components.The curl-explicit class has curl equal to its represented r; the regular-decomposition class has curl equal to curl z.Source: §5.2 · p. 20
  • Finite piecewise-affine, including ReLU, scalar potentials produce admissible H(curl) jumps: tangential components remain continuous while normal components may jump.The gradient carries the interface jumps in both 2D classes; the regular-decomposition class enriches the componentwise approach with this curl-free gradient.Source: §5.2 · pp. 20–21

5.3. Three dimensions.

  • In 3D, the regular-decomposition class grad ϕ + z lies in H(curl) for every realization, and dense approximations of its H1 components yield dense native classes.The decomposition is bounded and linear, has no separate harmonic part, and allows topological components to be absorbed into z.Source: §5.3 · pp. 21–22

5.4. Normal jumps and interface modules.

  • Across smooth interfaces, the scalar-potential kink carries the entire admissible normal jump, while the remainder has no jump; fixed or learned interface modules encode this mechanism.A prescribed jump j is represented by [[∂nϕ]] = j; with a fixed distance-function module, Lipschitz jump amplitudes can be attained and approximated by networks.Source: §5.4 · pp. 22–23

5.5. Relation to finite elements and componentwise networks.

  • In 2D, the curl-explicit neural class replaces Raviart–Thomas polynomials in the corresponding Nédélec representation; no such correspondence is claimed for the 3D class.In 3D, the H1 remainder replaces the Poincaré-operator component because curl cannot be prescribed freely.Source: §5.5 · pp. 23–24
  • Componentwise H1 networks are H(curl)-conforming but suppress normal jumps, whereas grad ϕθ + zθ permits finite-width normal jumps with correct tangential traces.Componentwise networks are analogous to vector Lagrange elements (C0)n, while the structured classes are analogous to Nédélec elements, exactly on a triangle in two dimensions.Source: §5.5 · p. 24

6. The FOSLS-deRhaNN method

FOSLS-deRhaNN combines conforming de Rham neural trial classes with the least-squares functional in the weak formulation’s natural spaces. Under stability and density conditions, its best-approximation error converges, while the computation uses numerical integration and optimization procedures not covered by convergence analysis.

6.1. FOSLS-deRhaNN approximation.

  • FOSLS-deRhaNN uses native classes for graph-space unknowns and minimizes the FOSLS functional in the natural spaces of the weak formulation.The trial realizations lie in their target spaces for every parameter value.Source: §6.1 · p. 24
  • Under FOSLS stability, the best-approximation error tends to zero as the suboptimality defect vanishes when the trial classes have dense unions.For second-order elliptic equations, the stated stability condition holds under unique solvability for right-hand sides in L2 and H−1.Source: §6.1 · pp. 24–25

6.2. Remarks on the trial classes.

  • For trial classes sharing hidden layers, the product-class approximation estimates apply with widths added, not at the same width.At a fixed width, the shared architecture is generally a proper subset of the product of independent scalar classes.Source: §6.2 · p. 25
  • Conformity alone does not ensure convergence: a proper closed subspace of the target space leaves a nonzero best-approximation error.On nonsmooth domains with Maxwell-type boundary conditions, an H1-conforming single-field class may fail to be dense in the intersection graph space.Source: §6.2 · p. 25
  • The first-order system assigns each variable its own continuity space, while intermediate-degree exterior-derivative equations require the prescribed g to satisfy dg = 0.An unconstrained trainable g can be built into the trial class only in top degree.Source: §6.2 · p. 25
  • A divergence equation can be built into the first H(div) class by setting q = f − B(z′), eliminating one network and one residual but tying q to z′.This is an option used for the generic transport level, not the main formulation.Source: §6.2 · p. 25

6.3. Computation.

  • The numerical examples use a linear least-squares solve for output coefficients and nonlinear optimization for remaining parameters, but analyze neither quadrature nor optimization convergence.The computed approximation depends on how the functional is integrated.Source: §6.3 · pp. 25–26
  • In computation, derivative residuals are integrated either on partitions cut along kink sets or through cellwise moments, with finer functional evaluation used to check iterates.The moment-based training functional is not generally a lower bound for the exact functional.Source: §6.3 · p. 26

7. Second-order elliptic equations

The section formulates discontinuous-coefficient elliptic problems as first-order systems with flux in H(div), then approximates the solution and flux using conforming neural classes. Its examples show low errors for both flux classes and demonstrate learned-interface approximation in dimensions four and eight.

7.1. Problem and first-order system.

  • The weak problem allows f1 in L2, f2 in L2, and Dirichlet data in H1/2, with point sources excluded for n ≥ 2.Writing the H^-1 right-hand side as f1 + div f2 accommodates discontinuous f2 and sources on curves; the flux unknown σ is not necessarily the physical flux when f2 jumps.Source: §7.1 · p. 27

7.2. Structured-flux formulation.

  • Both structured-flux classes are H(div)-conforming for every parameter value, with divergence q in the first class and div z in the second.Their fluxes are curl ψ + R2q + h and curl ψ + z; neither q nor discontinuous coefficients are differentiated in the residuals.Source: §7.2 · p. 28

7.3. Weakly imposed Dirichlet condition.

  • The Dirichlet condition can be imposed weakly by adding an H1/2 boundary residual to the least-squares functional.The boundary term is covered by continuous stability theory; an L2 boundary residual with a large weight is also described.Source: §7.3 · p. 29

7.4. Built-in variant: the PDE-balanced formulation.

  • The PDE-balanced class enforces the divergence equation identically, leaving only the constitutive residual while retaining error equivalence.It requires two scalar neural fields, v and ψ, plus finitely many harmonic coefficients; flux approximation reduces to H1 approximation of u and ψ* apart from the harmonic component.Source: §7.4 · pp. 29–30

7.5. Numerical examples.

  • After 90 minutes, the two classes give u errors of 0.28% and 0.30% and σ errors of 0.26% and 0.27%.The second run made 274 steps and the first 422; the first class's Jcheck fell from 4.1 · 10^-3 to 8.0 · 10^-4 and was still decreasing at the time limit.Source: §7.5 · p. 31
  • For the learned interface, the second class has smaller errors and mean interface distance at n=4, but larger errors and distance at n=8.At n=8, the first class's learned interface lies within 1.4 · 10^-4 of Γ and its errors are below 1%; the known-interface second class gives smaller u and σ errors by factors 1.5 and 2.7.Source: §7.5 · p. 32

8. Curl–curl problems

For curl–curl problems, the paper uses conforming neural approximations for the electric and auxiliary fields, with a least-squares functional equal to the energy-norm error. In the numerical example, the energy error is below 1.5% at every interface-information level and about 0.5% when the interface is known.

8. Curl–curl problems

  • The curl–curl least-squares functional is exactly the error in the energy norms, so no separate stability argument is needed.The method enforces conformity structurally, but neither curl–curl equation is built into the network.Source: §8 · pp. 32–33
  • Across coefficient interfaces, E and H have continuous tangential traces, while their normal components may jump.The normal jumps are represented by kinks in scalar potentials; the regular parts have no jumps.Source: §8 · p. 33

8.1. Numerical example.

  • The energy-norm error is below 1.5% at all levels and about 0.5% with the interface known.At Level II, all three initial networks move the learned zero set from the tilted starting plane to the interface.Source: §8.1 · p. 34

9. Linear transport

The section formulates linear transport with the flux as an H(div) unknown and applies two native flux classes to solutions with discontinuities. In the numerical examples, learned or generic network interfaces represent jumps without a transition layer.

9. Linear transport

  • The least-squares functional is the error in ||| · |||, but the paper claims no lower bound by the H(div; Ω) × L2(Ω) norm or L2 error estimate for v.The admissible trial pairs have square-integrable boundary traces; the native classes do not preserve a prescribed normal trace on the inflow boundary.Source: §9 · pp. 34–35
  • The flux classes are curl ψθ + R2qθ or curl ψθ + zθ, with the second class using the first-level divergence residual instead of an explicit q.The first class can build the divergence equation in at the second level, with q = f − γv; the second class cannot.Source: §9 · p. 35, §9.1 · p. 36
  • Across a characteristic with a jump in u, the flux jump is tangential and is carried by a kink of the potential; the divergence may jump as well.The section allows jumps in v and q through an interface variable or steps of generic-network units, while a continuous network approximates the jump by a transition.Source: §9 · p. 35

9.1. Numerical example.

  • The learned interface moves onto the characteristic within 1.5 · 10−5, and the computed solution resolves the jump rather than a transition layer.The first class's error is largest in a narrow downstream band along Γ, at about 3 · 10−4, apart from points between the learned curve and Γ.Source: §9.1 · p. 36, Figure 4 · p. 37
  • The second class gives smaller values in all reported quantities, by factors from about 2 to 14, and needs no ray evaluation.The first class's functional was still decreasing at the time limit; the second class completed 7586 iterations in 60 minutes, versus 1560 for the first.Source: §9.1 · p. 36
  • At generic level, the first class's learned step follows the exact jump with a relative error of 1.6%, while the second class's step has a relative error of 2.6%.For the second class, the tied zero set lies a mean distance of 7.0 · 10−4 from Γ, and the relative divergence error is 0.52%.Source: §9.1 · p. 36

10. Scalar conservation laws: an unknown moving interface

The paper treats scalar conservation laws with an independent native H(div) space-time flux, without supplying the shock location to the generic flux class. Its Burgers example resolves the moving shock within one to two training cells, and a consistency result identifies limits with weak solutions.

10. Scalar conservation laws: an unknown moving interface

  • The generic native flux class is used without a supplied shock location or interface module, replacing the H(div)-conforming finite-element flux in the flux least-squares formulation.The space-time flux has a continuous normal component across its kinks by construction; where τΘ = f(vθ), this is the Rankine–Hugoniot relation.Source: §10 · pp. 37, 40
  • Across a shock, the space-time flux has continuous normal component while its tangential components jump, making the shock an H(div) interface of unknown location.A ReLU potential can represent shocks as trained kinks; with a smooth activation, the shock becomes a layer that must be resolved by quadrature.Source: §10 · pp. 38–40
  • A divergence-free native flux class sets qη = 0, so the divergence residual vanishes and the least-squares functional reduces to its last three terms.For n = 1, this class is the space-time curl of one scalar network, τΘ = (∂xψθ, −∂tψθ).Source: §10 · p. 39
  • If the least-squares functional tends to zero and the trial pairs converge in L2 under the stated flux conditions, the limit is a weak solution with the inflow data.For a locally Lipschitz flux such as Burgers’ flux, the result assumes uniformly bounded L∞ norms of the trial and limit solutions.Source: §10 · p. 40

10.1. Numerical example.

  • Jcheck fell from 5.7 · 10−2 to 1.2 · 10−3, while the L1 error fell from 4.2% to 2.0%.The Burgers shock was resolved within one to two training cells without oscillations; its position was ahead of x = t/2 by at most 0.014 for t ≤ 0.6 and by 0.003 at t = 1.Source: §10.1 · p. 40

11. Concluding remarks

The constructed deRhaNN classes are conforming for every parameter value, with admissible jumps carried by potential kinks at finite width. Coupled with FOSLS, they are applied to elliptic, curl–curl, transport, and conservation-law examples.

11. Concluding remarks

  • The H(div) and H(curl) graph-space fields are assembled from ordinary neural representing functions using fixed de Rham operators.The classes are conforming for every parameter value, and potential kinks carry admissible interface jumps at finite width.Source: §11 · p. 40
  • The Burgers figure compares the predicted solution and flux-derived quantity with the exact solution at t = 0.25, 0.5, and 1.It also depicts the predicted solution on Q, the exact shock x = t/2, and potential kinks carrying tangential flux jumps of at least 2% of the exact jump.Source: Figure 7 · p. 41
  • Combined with first-order system least squares, the classes give the FOSLS-deRhaNN method.The elliptic and curl–curl examples retain natural graph-space stability; transport and conservation-law examples use the same native flux constructions for discontinuous solutions and moving interfaces.Source: §11 · p. 41
Loading 2610.08016v1…