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Latent Inversion of Material Coefficients from Boundary Data via Finite Tests and Neural Surrogates

Erik Burman, Mats G. Larson, Karl Larsson, Jonatan Vallin

arXiv:2609.17137v1math.NAcs.NA

TL;DR

Boundary-based coefficient recovery is severely unstable, while full operator data and repeated PDE solves limit practical reconstruction. The paper uses low-dimensional latent families and finitely many boundary tests, establishing local stability that transfers to finite elements and surrogates. Two-dimensional experiments illustrate reconstruction and reduced online cost, with further improvement from optional FEM refinement.

  • Problem

    Recovering a coefficient from boundary data is severely unstable, and full NtD operator data and repeated PDE solves pose practical limitations.

    Source: §1 · pp. 1–2

  • Method

    The paper restricts coefficients to a low-dimensional latent family and uses finitely many Neumann excitations, transferring local stability through finite element and surrogate maps.

    Source: §1 · p. 2, §4.5 · p. 23

  • Results

    At most m excitations yield local Lipschitz stability under tangent-space injectivity; two-dimensional experiments illustrate recovery of principal geometric features and further error reduction with FEM refinement.

    Source: §7 · p. 45

  • Takeaways & Limitations

    Latent-space reconstruction can reduce online evaluation cost, and FEM-based refinement in the same latent space can further reduce boundary misfit and coefficient error.

    Source: §7 · p. 45

    The numerical tests do not certify the required nondegeneracy or approximation conditions.

    Source: §6.4 · p. 30

Abstract

from arXiv · show

We reconstruct a spatially varying material coefficient in a scalar elliptic equation from finitely many boundary excitations, each producing a full Dirichlet trace. To mitigate the ill-posedness and the cost of repeated PDE solves, we restrict the coefficient to a low-dimensional family, specified analytically or learned from samples, and solve the inverse problem in its latent coordinates. We consider a \(C^1\) parametrization with \(m\) latent coordinates and full-rank derivative at a reference point. If the continuous linearized Neumann-to-Dirichlet map is injective on the corresponding tangent space, at most \(m\) excitations suffice for local injectivity and Lipschitz stability. Convergence of the coefficient sensitivities then transfers this stability to conforming finite element discretizations. For sufficiently fine meshes, the stability constant and neighborhood can be chosen independently of the mesh size. Under a local residual-comparison condition, uniform accuracy of the surrogate forward map yields coefficient-error bounds separating representation error, data noise, finite element error, and surrogate error. Derivative accuracy additionally preserves the surrogate's own local stability. All stability statements are local to a reference coefficient. Two-dimensional numerical experiments combine analytic and learned representations of inclusions and crack-like coefficients with neural forward surrogates. They illustrate latent-space reconstruction, reduced online cost, and further improvement from optional FEM-based refinement.

1 Introduction

The introduction frames coefficient recovery as severely unstable from boundary data and motivates reducing both the coefficient space and the number of experiments. It presents local latent-space stability, its transfer to finite element and surrogate maps, and numerical illustrations.

1 Introduction

  • The Calderón problem reconstructs conductivity from boundary measurements, but recovery is severely unstable and full NtD data and repeated PDE solves are practical limitations.These constraints motivate reducing both the coefficient space and the number of boundary experiments.Source: §1 · pp. 1–2
  • If admissible Neumann tests separate every nonzero latent tangent perturbation, at most m tests yield local injectivity and Lipschitz stability near the reference coefficient.The resulting coefficient difference is bounded by a constant times the aggregate H1/2 boundary-trace discrepancy.Source: §1 · p. 2
  • Under suitable regularity and dimensional assumptions, complex geometrical optics solutions verify the required separation, with an elementary 2D example for affine families near a constant coefficient.Source: §1 · p. 2
  • Convergent coefficient sensitivities transfer stability to sufficiently fine conforming finite element meshes, while uniform surrogate value accuracy bounds reconstruction error and derivative accuracy preserves surrogate stability.The surrogate bounds require residual comparison; the approximation conditions are explicit hypotheses, not certified by training.Source: §1 · p. 3
  • The theory is local within the latent family; experiments with analytic and learned inclusions and crack-like coefficients illustrate reconstruction, reduced online cost, and optional FEM refinement.For targets outside the family, the error bound adds representation error relative to a comparison coefficient in the local stability set.Source: §1 · p. 3
  • Prior work establishes finite-measurement and latent-representation approaches, while this paper transfers local finite-test stability through continuous, finite element, and surrogate maps by controlling coefficient sensitivities.Source: §1 · p. 4

2 Mathematical and Computational Framework

The framework formulates boundary-trace measurements as an inverse problem, then restricts coefficient recovery to a low-dimensional latent family. It introduces conforming finite element and neural-surrogate forward maps for that reduced problem.

2.1 Continuous Forward and Inverse Problems

  • The inverse problem uses Dirichlet traces from prescribed Neumann experiments to fit an unknown coefficient, with the search restricted to a latent family to incorporate prior structure and reduce unknowns.If the exact coefficient lies outside that family, the reduced problem seeks its best fit within the family.Source: §2.1 · pp. 6–7
  • The representation map sends a low-dimensional latent variable to a material coefficient and may be specified analytically or learned from samples; it is fixed during inversion.The reduced inverse problem estimates the latent variable and maps it back to a coefficient.Source: §2.1 · pp. 6–7

2.2 Finite Element Method

  • The finite element method uses a conforming partition and standard conforming spaces to define discrete Neumann responses for the selected boundary tests.The synthetic discrete datum is model-generated using the exact coefficient, unlike independently measured data.Source: §2.2 · pp. 7–8

2.3 Neural Surrogate Approximation

  • The neural surrogate approximates the discrete finite-test response map directly from latent inputs, and the surrogate-based inverse problem keeps its optimization variable in latent space.Training targets are the discrete responses at latent training points, represented in coordinates of the finite-dimensional observation space.Source: §2.3 · pp. 8–9

3 Continuous Theory for Latent Inversion with Finite Tests

The continuous theory differentiates the coefficient-to-boundary map, uses separation on the latent tangent space to select finite Neumann tests, and establishes local stability around z†.

3 Continuous Theory for Latent Inversion with Finite Tests

  • The analysis differentiates the NtD map and composes its sensitivity with the latent map before applying separation to the tangent space.The tangent space is the range of DΦ(z†); no global embedded-manifold structure is assumed.Source: §3 · pp. 9–10, §3.3 · p. 11

3.1 Differentiability of the Continuous NtD Map

  • The continuous solution map is continuously Fréchet differentiable near uniformly positive coefficients, providing the sensitivities used in the inverse analysis.The resulting boundary map is also continuously Fréchet differentiable.Source: §3.1 · p. 10

3.2 Differentiability of the Latent Forward Map

  • The latent reduced forward map is C1, with its derivative obtained by composing the coefficient sensitivity with the latent parametrization.Its components are boundary responses for the selected Neumann tests.Source: §3.2 · p. 11

3.3 Linearized Injectivity and Finite Neumann Tests

  • Gradient-product separation on the latent tangent space implies linearized NtD injectivity, allowing at most m tests with a positive definite cumulative Gram matrix.Each test is a Neumann excitation producing a full Dirichlet trace.Source: §3.3 · pp. 12–13
  • A positive definite selected-test Gram matrix yields local injectivity and Lipschitz stability in latent and coefficient variables near z†.The result holds on a convex neighborhood of the reference latent point.Source: §3.3 · p. 13
  • The finite-test selection is existential and reference-dependent, so it does not certify one prescribed test family that works throughout the latent model.A CGO-based sufficient condition for separation is given for specified settings, including a two-dimensional constant-reference example.Source: §3.3 · pp. 12, 14

4 Discrete and Surrogate Theory for Latent Forward Maps

This section extends local latent stability from continuous finite-test maps to finite-element and neural-surrogate residual maps, including boundary L2 observations. Mesh-independent stability and reconstruction bounds are established only under the stated sensitivity, approximation, locality, and residual-comparison conditions.

4.1 Discrete Well-Posedness and Differentiability

  • Under the discrete well-posedness assumption, the finite element state is unique and its latent-coordinate state map is C1, with uniformly controlled sensitivities.Uniform coefficient bounds and coercivity support the existence and sensitivity estimates.Source: §4.1 · p. 15

4.2 Derivative Error Estimates

  • Finite element state and sensitivity estimates yield convergence of the finite-test map and its derivative, including operator-norm convergence of the discrete derivative at the reference point.The estimates assume coefficient values remain admissible and, for sensitivity estimates, a C1 coefficient parametrization; the approximation property gives convergence as the mesh is refined.Source: §4.2 · pp. 15–17

4.3 Positivity of the Discrete Gram Matrix and Local Stability

  • For sufficiently fine meshes, the discrete finite-test map is locally injective and Lipschitz stable on a neighborhood and with a constant independent of mesh size.The continuous cumulative Gram matrix must be positive definite, and the finite element spaces must satisfy the standard approximation property.Source: §4.3 · p. 18

4.4 Stability under Neural-Surrogate Approximation

  • Small C1 perturbations preserve local injectivity and stability, whereas uniform value accuracy alone suffices for surrogate-based material-error bounds under residual comparison.The value-accuracy bound does not require derivative accuracy or surrogate injectivity; the comparison condition holds for a global residual minimizer on the local compact set, not necessarily an arbitrary iterate.Source: §4.4 · pp. 20–22
  • The neural C1-convergence hypothesis is not established for trained networks, and training or validation mean squared errors do not certify the uniform surrogate-error bound.Thus the stated stability and error conclusions are conditional on approximation assumptions rather than guarantees about the trained networks or optimization algorithm.Source: §4.4 · pp. 21–22
  • The residual bound is E_h + C_F e_rep + ε, but applies locally only when both latent points lie in K and residual comparison holds.The terms separate discretization, representation, and surrogate contributions.Source: §4.4 · p. 22

4.5 Stability in the Boundary L2-Norm

  • Boundary L2 observations retain local stability on a neighborhood independent of h for sufficiently fine meshes.The stability constant is also independent of h.Source: §4.5 · p. 23
  • A C1 surrogate inherits local stability if its derivative error is at most 1/(2c_d,L^(0)) on the set.The derivative error is measured in the operator norm from R^m to the boundary L2 observation space.Source: §4.5 · p. 23
  • The conclusions are local to z†; mesh-independent bounds require finite-element approximation, and surrogate stability remains conditional on coordinate-space C1 approximation.The analysis provides no guarantee for a particular network or training procedure, mesh-uniform width bound, or joint relation between M and h.Source: §4.5 · p. 23

5 Material Representation Maps

The paper describes analytical and learned maps from low-dimensional latent coordinates to admissible material coefficients. Its local stability results require each representation to meet regularity, rank, and test-separation conditions.

5 Material Representation Maps

  • Each representation must satisfy admissibility, C1 regularity, full rank at the reference point, and positive definiteness of the selected tests' cumulative Gram matrix.The selected excitations depend on the coefficient family and reference coefficient.Source: §5 · pp. 23–24, Analytical Parametrization · p. 25

Analytical Parametrization.

  • Analytical parametrizations use interpretable parameters directly as latent variables, while sharp-interface motion may fail the required C1 regularity into L∞(Ω).The binary analytical ellipse is used as a numerical illustration outside Assumption (A1).Source: Analytical Parametrization · p. 24
  • Linear representations combine a baseline coefficient with modes, and satisfy the full-rank condition exactly when those modes are linearly independent.The latent set must also keep represented coefficients admissible.Source: Analytical Parametrization · p. 24
  • At most m Neumann excitations yield local Lipschitz stability near a constant reference for sufficiently small affine families in two dimensions.This conclusion assumes linearly independent modes and the discrete results additionally require the approximation assumption of Theorem 4.1.Source: Analytical Parametrization · p. 24
  • A learned decoder maps latent coordinates to normalized coefficient fields, then rescales them into the admissible coefficient range.The decoder is trained offline and fixed during inversion; C1 regularity and full rank at the reference point must be checked rather than presumed from training.Source: Analytical Parametrization · p. 25

6 Numerical Experiments

The numerical experiments use pixelwise coefficient fields and finite element boundary responses to illustrate the reconstruction pipeline. They include a variational autoencoder representation, while the experiments are not intended to verify the theoretical separation condition.

6 Numerical Experiments

  • The experiments illustrate the reconstruction pipeline and computational cost, rather than numerically verifying the separation condition or its sufficient CGO criterion.All computations are two-dimensional and use pixelwise piecewise-constant coefficient fields.Source: §6 · p. 25

6.1 Problem Setup

  • Coefficients are represented as 64 × 64 pixel images on a uniform grid over the square domain [−1, 1]².The ambient pixel space has dimension 4,096.Source: §6.1 · p. 26
  • Each boundary excitation uses a smooth, zero-mean Fourier-mode Neumann pattern, and its FEM solution contributes a trace vector containing values at all boundary nodes.The modes are orthogonal in L2(∂Ω) and probe low-to-moderate frequencies.Source: §6.1 · p. 26
  • Coefficient samples for the VAE, surrogate data, and reconstruction targets are independent draws from the same family, with targets excluded from training and validation sets.The VAE and surrogate use separate training and validation splits.Source: §6.1 · p. 26

6.2 Variational Autoencoder for Material Representation

  • The VAE encodes normalized coefficient images into a latent distribution, samples a latent vector, and decodes it into a reconstructed image; prior samples generate new images.The latent dimension m is much smaller than the 4,096-pixel ambient dimension.Source: §6.2 · p. 27, Figure 4 · p. 28
  • Adding the positive offset amin to normalized decoder outputs keeps physical coefficients between amin and amax and within the admissible pixel coefficient set.Plots omit amin and display only the normalized field.Source: §6.2 · p. 27
  • The VAE uses convolutional encoder and decoder blocks, and trains with pixelwise reconstruction, edge-aware gradient matching, and Gaussian latent regularization.The decoder progressively upsamples its latent representation and restricts reconstructed pixel values to [0, 1].Source: §6.2 · pp. 28–29

6.3 Neural Forward Surrogate

The neural surrogate is trained to approximate finite-test latent boundary responses, reducing the need for repeated FEM solves during online inversion. Its training pairs come from either a pretrained VAE representation or an analytical parametrization.

6.3 Neural Forward Surrogate

  • The surrogate approximates the latent finite-test forward map to avoid repeated expensive finite element solves during online inversion.Its accuracy is assessed primarily through boundary-data misfit.Source: §6.3 · p. 29
  • For a learned representation, encoded coefficient samples supply latent inputs paired with their FEM boundary-response outputs for surrogate training.The training targets are evaluations of the discrete finite-test map.Source: §6.3 · p. 29
  • The surrogate network is overparameterized relative to the training-pair count and fitted by minimizing mean squared prediction error.The supplied passages do not specify the hidden-unit count.Source: §6.3 · p. 29
  • For analytical parametrizations, the method samples parameter values, maps them to coefficients, computes their discrete boundary responses, and trains on those pairs.The surrogate uses either ReLU or ELU activations.Source: §6.3 · p. 29

6.4 Inverse Problem Objectives

The section defines a locally stable boundary-data objective for reconstruction and describes its finite element implementation. The numerical tests use elliptic-inclusion and crack-like coefficient families.

6.4 Inverse Problem Objectives

  • On sufficiently fine meshes, local stability holds in the boundary L2 norm with a mesh-independent constant, but the numerical tests do not certify its required conditions.The target is approximated by minimizing the surrogate objective.Source: §6.4 · p. 30
  • The surrogate and FEM objectives use boundary mass-matrix quadratic forms of predicted-minus-observed nodal residuals, including surrogate boundary-mean error.The FEM objective is differentiated with respect to latent variables through the finite element solver using JAX automatic differentiation.Source: §6.4 · p. 30
  • The experiments consider piecewise-constant elliptic inclusions and localized crack-like defects in smooth random backgrounds.Both are coefficient families used in the numerical tests.Source: §6.4 · p. 30

6.5 Elliptic Inclusions

The ellipse experiments compare an interpretable analytical parametrization with a learned VAE representation, using surrogate-based latent reconstruction. FEM refinement improves both data fit and coefficient reconstruction in the displayed examples.

6.5 Elliptic Inclusions

  • The analytical ellipse family uses five geometric parameters, and the experiments use six boundary excitations.The parameters specify the ellipse center, principal semi-axes, and rotation; the family also supports comparison with a learned representation.Source: §6.5 · pp. 30–31

6.5.1 Analytical Parametrization

  • For two analytical-ellipse targets, reconstructions closely match the coefficients, with discrepancies mainly near ellipse boundaries and similar surrogate and FEM objective trends.The optimization minimizes the surrogate objective over the five-dimensional parameter space.Source: §6.5.1 · p. 32

6.5.2 VAE Parametrization

  • When the analytical parametrization is unavailable, a VAE learns a six-dimensional representation from coefficient samples, and a surrogate maps latent vectors to boundary responses.New fields can be generated by decoding samples from the standard Gaussian prior; displayed samples resemble the training data geometrically.Source: §6.5.2 · pp. 32, 34
  • For displayed learned-ellipse reconstructions, errors concentrate near ellipse boundaries; the decoder can produce intermediate pixel values rather than binary inclusions.Along the shown trajectories, the surrogate objective follows the overall trend of the FEM objective.Source: §6.5.2 · p. 35
  • For four ellipse targets, FEM-based Newton-CG refinement decreases both the FEM objective and coefficient L2 error after surrogate-based initialization.Figure 13 marks the transition after 150 surrogate-gradient iterations; the refined reconstructions are shown at iteration 200.Source: Figure 13 · p. 37, §6.5.2 · p. 38

6.6 Crack-Like Defects in Heterogeneous Backgrounds

The crack experiments combine a random heterogeneous background with localized crack defects, represent the family using a VAE, and reconstruct in latent space with a neural surrogate. FEM refinement improves both data fit and coefficient reconstruction in the displayed examples.

6.6 Crack-Like Defects in Heterogeneous Backgrounds

  • The coefficient family combines a spatially correlated random background with single random-polyline cracks, whose pixels replace the background with value 1 before the positive offset.The background uses a truncated Karhunen–Loève expansion of a squared-exponential covariance; cracks are grown in line segments until both ends reach the boundary.Source: §6.6 · p. 38

6.6.1 VAE Parametrization

  • The learned crack representation uses an eight-dimensional latent space, ReLU activations, and ten boundary excitations.The VAE samples reproduce characteristic geometric and background features of the training data.Source: §6.6.1 · pp. 38–39
  • For displayed targets, the learned representation and surrogate recover dominant crack geometry and main background features, with errors near crack boundaries and in parts of the background.Softened transitions near cracks are empirical reconstruction features, not a necessary consequence of decoder continuity.Source: §6.6.1 · p. 40
  • Along the displayed crack-reconstruction trajectories, the surrogate objective follows the overall trend of the FEM objective.The figures also show coefficient L2-error histories.Source: §6.6.1 · pp. 40–41
  • For four crack targets, FEM-based Newton-CG refinement rapidly decreases the FEM objective and improves coefficient reconstruction after surrogate-based initialization.The refinement uses 50 FEM iterations after 150 surrogate iterations.Source: §6.6.1 · p. 41

6.7 Computational Cost

The cost evaluation compares surrogate and FEM forward evaluations and reports representative reconstruction and training times. The surrogate is faster per forward evaluation, while FEM refinement dominates the refined pipeline's runtime.

6.7 Computational Cost

  • For this implementation and discretization, surrogate evaluation is approximately 109 times faster than the complete decoder–FEM pipeline on CPU and approximately a factor of 10 faster on GPU.The table reports average CPU and GPU wall-clock times per forward evaluation, computed from 100 evaluations, each producing boundary responses for L = 10 boundary excitations.Source: §6.7 · p. 42, Table 1 · p. 42
  • For the crack experiment, offline training took approximately 9 minutes 32 seconds for the surrogate and approximately 2 hours 34 minutes for the VAE.The training datasets used 4,500 surrogate training pairs and 9,000 VAE training fields.Source: §6.7 · p. 42, Table 2 · p. 44
  • A representative surrogate-only reconstruction takes approximately 5 seconds; the refined pipeline takes approximately 140 seconds, including 8 seconds for the surrogate stage and 132 seconds for FEM refinement.The reported times depend on the target coefficient and initialization.Source: §6.7 · p. 44, Table 3 · p. 44

7 Conclusions

The paper establishes local latent-space stability from finitely many boundary tests and transfers it to fine finite element meshes. Surrogate error bounds and two-dimensional experiments illustrate the framework, with explicit limits on the guarantees and experimental scope.

7 Conclusions

  • If the linearized boundary map is injective on the m-dimensional latent tangent space, at most m excitations give local Lipschitz stability, preserved on sufficiently fine meshes with mesh-independent constants and neighborhoods.The argument also applies to the boundary L2-norm.Source: §7 · p. 45
  • Uniform surrogate value accuracy yields a local coefficient-error bound separating representation, data, finite element, and surrogate errors, while derivative accuracy preserves surrogate local stability.These results require residual comparison in the local stability set; tests and neighborhoods may depend on the reference coefficient, and optimizer convergence is not guaranteed.Source: §7 · p. 45
  • Two-dimensional experiments recover principal features of analytical and learned inclusions and crack-like coefficients; latent FEM refinement further reduces misfit and coefficient error, while the surrogate is cheaper to evaluate.The experiments do not verify the separation or surrogate-approximation hypotheses, and moving binary-interface and ReLU representations fall outside the global C1 assumptions.Source: §7 · p. 45

A CGO Verification of the Separation Condition

The appendix verifies the separation condition using complex geometrical optics solutions under specified smoothness and dimensional assumptions. The proof converts conductivity solutions to a Schrödinger form, extracts perturbation information from CGO products, and identifies a smaller sufficient richness condition on the boundary tests.

A CGO Verification of the Separation Condition

  • For d ≥3 and a bounded, connected domain with smooth boundary, the CGO argument verifies the separation condition, furnishing a sufficient criterion for the abstract injectivity argument.The dimensional and smoothness assumptions are not imposed on the two-dimensional numerical experiments; the proposition follows.Source: A CGO Verification of the Separation… · pp. 45, 49
  • A Liouville transform recasts conductivity solutions as Schrödinger solutions, whose CGO products yield distributional information about the perturbation.The conclusion establishes the separation condition for every latent tangent space.Source: A CGO Verification of the Separation… · pp. 46, 49
  • The full compatible Neumann test class can be weakened to a class containing the real and imaginary parts of the CGO-generated conormal traces.The proof also notes that complex-valued CGO solutions are handled by applying the real bilinear identity to their real and imaginary parts.Source: A CGO Verification of the Separation… · pp. 47, 49
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