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Stabilizing and Solving Unique Continuation Problems by Parameterizing Data and Learning Finite Element Solution Operators

T0 review · 2 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read A stabilized finite element scheme makes unique continuation first-order accurate when boundary data are low-dimensional, and a learned decoder-plus-operator network extends the idea to nonlinear PDEs.

desk verdict The linear reduced-order error analysis is solid and new; the nonlinear experiments are self-referential and need to be re-run against true FE solutions before the practical claims can be trusted. read the letter →

arxiv 2412.04409 v3 pith:TWND27QP submitted 2024-12-05 math.NA cs.LGcs.NA

Erik Burman , Mats G. Larson , Karl Larsson , Carl Lundholm This is my paper · ORCID
classification math.NAcs.LGcs.NA MSC 65N3065N2135R30
keywords uniquecontinuationinverseproblemsstabilizedfiniteelementmethodproperorthogonaldecompositionautoencoderoperatorlearningreducedordermodelill-posed
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to establish that the severely ill-posed problem of reconstructing a PDE solution from a bulk interior measurement, when the missing boundary data are known to lie on a low-dimensional set, can be made stably and optimally convergent. It proves for a linear Laplace model that a stabilized projection onto the span of finitely many precomputed modes converges in $H^1$ with error $O(h)$ times the data norm, independent of the near-instability of the continuous inverse problem. It then builds a composition of an autoencoder decoder and a learned finite element solution operator, so the same inverse problem becomes an unconstrained minimization over a latent space. If the method is right, it offers a template for turning abundant historical boundary observations into a stable prior for otherwise ill-posed data-assimilation problems.

What carries the argument

The load-bearing object is the stabilized bilinear form $m_h(v,w) = (v,w)_\omega + h^{-1}(v - I\hat v, w - I\hat w)_{\partial\Omega} + h(\nabla_T(v - I\hat v), \nabla_T(w - I\hat w))_{\partial\Omega} + \sum_{F\in\mathcal{F}_h} h([\nabla v],[\nabla w])_F$, where $I$ maps coefficient vectors onto the continuous harmonic modes and $I_h$ onto their Nitsche-computed finite element approximations. The first stabilization term controls the distance of the discrete solution to the finite-dimensional trace set $G$ in the $H^{1/2}$ norm, and the jump-penalty term supplies enough coercivity that the coefficient bound $\|\hat v\| \lesssim \|v\|_{m_h}$ holds uniformly in $h$, which is exactly what the plain $L^2(\omega)$ projection lacks. This stabilizes the reduction step in the linear theory. In the nonlinear pipeline, the same idea is replaced by a learned composition of decoder and operator network, so the constraint 'boundary data in $G$' is enforced structurally rather than by a penalty.

What would settle it

Pick a target solution whose boundary trace is a known linear combination of POD modes with coefficients inside the training support, generate its bulk observation $u_0$ in $\omega$, and run the latent-space minimization; the linear theory predicts error $O(h\|u_0\|_\omega + \delta)$, so if reconstruction error is large even when the decoder's reconstruction error $\delta$ is tiny, the learned decoder-operator composition is the point of failure. A more direct check is to compute the decoder's boundary reconstruction error for the specific target trace and the operator network's output error; the paper's claim is stable reconstruction follows once those errors are small, so a target with small decoder error and small operator error that still yields large interior error would falsify the method's premise.

Watch

Extended reading notes

Core claim

On its own terms, the paper's central result is Proposition 3.2: for boundary data drawn from an $N$-dimensional set $G$, the stabilized reduced-order projection $u_{N,h}$ defined by $m_h(u_{N,h}, v) = (u_0, v)_\omega$ satisfies $\|u_N - u_{N,h}\|_{H^1(\Omega)} \lesssim h\|u_0\|_\omega$, with constants growing with $N$ but not with the mesh size. The mechanism is a stabilized bilinear form that adds a tangential boundary control and a face-jump penalty, which restores the discrete stability that the naive Galerkin projection lacks because finite element functions do not satisfy unique continuation. The paper further claims that in practice the dimension reduction can be learned: POD gives a linear basis of collective boundary data, an autoencoder finds a nonlinear latent parametrization, and an operator network approximates the FE solution map from coefficients to degrees of freedom, so the inverse problem is solved by optimizing over the latent space. The nonlinear examples show the composition reconstructs the reference solution, and that a decoder trained on unperturbed data outperforms one trained on noisy data.

Load-bearing premise

Everything downstream depends on the true missing boundary trace lying inside, or within a small distance $\delta$ of, the low-dimensional set $G$ that the collective data and the autoencoder manage to capture; if the decoder cannot represent the target trace, the latent-space objective has no solution near the true field and the logarithmic ill-posedness of unique continuation returns.

Editorial extensions

If this is right

  • Linear unique continuation with finite-dimensional boundary trace is solved with optimal first-order $H^1$ accuracy; the constant grows exponentially in $N$, so the benefit of dimension reduction is quantitative, not just computational.
  • Using collective data to identify a lower-dimensional manifold improves stability from logarithmic to linear under the representability assumption, so abundant historical data can rescue an otherwise ill-posed reconstruction problem.
  • Autoencoders can find a lower latent dimension than PCA when the boundary-data manifold is nonlinear, as shown in the quadratic and Gaussian examples, giving a smaller optimization space for the inverse problem.
  • Composing the decoder with the operator network yields a differentiable mapping from latent variables to approximate FE solutions, so the inverse problem reduces to unconstrained minimization over the latent space.
  • A decoder trained on clean data reconstructs more accurately than one trained on noisy data, indicating that the quality of the collective data used for training the reduction map matters for the final reconstruction.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The theoretical guarantee stops at the linear reduced-order projection; extending Proposition 3.2 to the nonlinear learned pipeline would require a bound linking autoencoder and operator training errors to the reconstruction error, which the paper does not provide.
  • A testable consequence: for targets whose boundary trace lies in $G$ but is poorly reconstructible by the decoder, the method should fail even though the linear theory would succeed; comparing the decoder-free coefficient-space optimization with latent-space optimization on the same reference would isolate the representability loss.
  • The stabilization analysis suggests an alternative design for the learned pipeline: instead of decoding to coefficients and then applying the operator network, one could train the operator network on stabilized FE solutions generated by $m_h$, which might transfer the uniform-in-$h$ stability to the nonlinear case.
  • Because the stability constant grows exponentially with the latent dimension, the autoencoder's success in finding low-dimensional structure is not just a computational convenience but the main driver of accuracy; measuring the effective latent dimension from reconstruction-error plateaus could be used as a model-selection criterion.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper develops a computational framework for the unique continuation inverse problem in which, in addition to a bulk measurement u0 in a subdomain ω, one has access to a large set of boundary data GS from a low-dimensional family G. The authors combine POD and autoencoders to parametrize the boundary data in a latent space, train an operator network that approximates the FE solution map from boundary coefficients to finite element degrees of freedom, and then solve the inverse problem by unconstrained optimization in the latent space. For a linear model problem, they prove H1 error estimates for a stabilized reduced-order projection onto a space spanned by approximately computed harmonic basis functions, with first-order convergence in the mesh size h. For nonlinear problems, they present numerical demonstrations for a nonlinear diffusion-type operator with linear and nonlinear boundary-data models.

Significance. The linear theory is the main rigorous contribution: Proposition 3.2 appears to provide a clean O(h) H1 error bound for the stabilized reduced-order projection, extending earlier finite-dimensional-trace unique continuation results to a practical reduced-order setting. The paper also gives a concrete and reproducible numerical pipeline built on publicly available code, and it is honest about several limitations, including the fact that the nonlinear inverse tests use network-generated reference solutions. The practical claim that the ML framework solves the nonlinear inverse problem (2.5) is plausible but is not established by the current experiments, because the validation is partly self-referential and no error bound links the autoencoder/operator-network approximation to the inverse reconstruction error.

major comments (2)
  1. [Section 5.3, Figures 7 and 11–13, Table 3] The nonlinear inverse problems are validated against a reference solution generated by the same operator network used in the reconstruction: the reference is the network output for the input p14 = 10, all other pn = 0, and the data u0 is produced from that output. The authors explicitly acknowledge in the final remarks of Section 5.3 that 'taking some output of the method under consideration as the reference solution, as is done here, is typically not a proper choice since it is too idealized.' This is a load-bearing weakness: the optimizer only needs to invert the operator network, not to find a true FE solution of the nonlinear PDE, so the numerical results in Figures 7 and 11–13 do not demonstrate that the composed map (ϕu,N,h ◦ ϕde) solves the original inverse problem (2.5). The nonlinear experiments should be repeated with reference solutions computed by an independent FE solver and with errors measured against those solutions.
  2. [Eq. (4.12) and Section 4.1] The latent-space formulation assumes that the POD coefficients of the true boundary data lie in the image of the decoder, or at least within a small δ-neighborhood of it. The paper provides no bound linking the autoencoder reconstruction error and the operator-network approximation error to the reconstruction error of the inverse problem. If the target boundary trace is not representable by the decoder, the objective (4.12) has no minimizer near the true field and the logarithmic ill-posedness of unique continuation returns. This representability is an unverified engineering assumption, and Table 3 shows that it can fail substantially (the 'sd = 0.15' decoder gives MSE 1.99e-2 on ω). A numerical test with a reference solution generated from boundary data outside the decoder's training distribution, or an error decomposition along the lines of data approximation + autoencoder error + operator-learning error + optimization error, is needed to support the nonlinear practical claims.
minor comments (5)
  1. [Eq. (3.28)] The first inequality in the chain should read ∥bv∥_{R^N} ≲ ∥Ibv∥_ω rather than ∥Ibv∥^2_ω; as written the displayed chain is dimensionally inconsistent and does not match the subsequent argument.
  2. [Section 3.2] The inequality attributed to 'Galigliardo-Nirenberg' should be 'Gagliardo-Nirenberg'.
  3. [Eq. (3.52)] The statement that the constant in Proposition 3.2 depends 'on higher order Sobolev spaces of g' is vague; the authors should state the precise norm on g and the explicit dependence on N and λ_min, since this constant is the quantity that controls the ill-posedness of the problem.
  4. [Eq. (3.24)] The term I b v in the boundary stabilization uses the coefficient extraction operator b·, but the domain and action of b· are only defined implicitly after (3.13); this should be stated explicitly before the stabilized form is introduced.
  5. [Figures 7 and 11–13] Since the reference solution uses p14 = 10 while the operator network was trained with inputs from N(0, 0.09), it would be useful to report the training distribution's range for p14 and to explain why this out-of-distribution input was chosen for the reference.

Circularity Check

1 steps flagged · score 4.0 of 10

Nonlinear validation is self-referential: the reference solution is generated by the same operator network used in the reconstruction objective, so the numerical tests demonstrate self-consistency rather than PDE accuracy; the linear theory in Section 3 is not circular.

  1. fitted input called prediction [Section 5.3, paragraphs 'Combining Operator Network with Decoder' and final remarks; Figure 7 and Figures 11–13 captions; Table 3.]
    "The last frame shows ω and the reference solution used for the data which was obtained by taking p14 = 10 and all other pn’s = 0. ... As final remarks we point out that taking some output of the method under consideration as the reference solution, as is done here, is typically not a proper choice since it is too idealized."

    In every nonlinear inverse demonstration, the “reference solution” is the output of the trained operator network ϕu,N,h at a chosen coefficient vector (p14=10). The reconstruction minimizes (4.12), whose forward map is the same ϕu,N,h composed with a decoder. The target u0 is therefore, up to the decoder's representability, an element of the range of the model being optimized. A successful MSE fit measures whether the optimizer can find a latent or coefficient value reproducing the model's own output on ω; it does not measure error against an actual finite element solution of P(v)=0. The paper's own final remark concedes this choice is 'too idealized.'

full rationale

The linear analysis in Section 3 is self-contained and not circular. Proposition 3.2's error bound is proved from the stabilized projection (3.37), the discrete stability Lemma 3.1, Galerkin orthogonality, and a standard duality argument; no fitted parameter or prior self-citation is used as the load-bearing premise. Citations to [12] and [30] are contextual rather than load-bearing: the finite-dimensional-trace stability is re-proved in the text, and the operator-learning method is described rather than justified by a uniqueness theorem. The circularity is confined to the nonlinear validation protocol. The paper explicitly flags in Section 5.3 that 'taking some output of the method under consideration as the reference solution, as is done here, is typically not a proper choice,' and this admission is weighed in the verdict. In addition, the reference coefficient p14=10 lies far outside the training distribution N(0,0.09) used in Section 5.2, so the self-consistent test cannot detect out-of-distribution forward-model error. Because the central mathematical result remains independent while the nonlinear numerical claims reduce to self-consistency by construction, the score is 4 rather than 6–8.

Assumptions & free parameters 6 free parameters · 6 assumptions · 1 invented entities

The linear theorem relies on standard unique continuation and elliptic regularity plus the finite-dimensional trace assumption. The nonlinear ML pipeline adds unproved assumptions: representability of the target by the learned latent decoder, accuracy of the learned solution operator, and coverage of the collective data. The numerical validation is weakened by using network outputs as reference solutions.

free parameters (6)
  • POD truncation dimension N = 9 or 21 in experiments; chosen from significant singular value count
    Selects how many boundary modes are kept. The theoretical stability constant grows exponentially in N, so this hand-chosen truncation controls both stability and accuracy; no automatic rule is given.
  • Autoencoder latent dimension n_Z = 9 for Gaussian (nX,L)=(3,7) example
    Selected by trial and error from reconstruction-error plateaus in Figure 10c. It determines the dimension of the inverse optimization and is not derived from theory.
  • Operator network hidden width X = 64, 256, 512, 1024 for the four meshes
    Chosen by trial and error using a zero-energy check. Width trades approximation quality against training feasibility; no principled selection criterion.
  • Training hyperparameters (batch size, iterations, learning rate schedule) = batch 32/64/96; 1e6 Adam iterations; lr 1e-4 halved every 250k
    Hand-tuned. The paper states hyperparameters were obtained by trial and error, and the 244x244 case required randomly subsampling elements to avoid NaN loss.
  • Inverse optimization settings = Adam step size 0.1; convergence when 2-3 significant MSE digits stabilize
    Chosen by testing. The convergence criterion is based on visual inspection and significant-digit changes, not a statistical rule.
  • Reference solution coefficient p14 = 10
    Hand-picked synthetic coefficient for the nonlinear examples. Reported MSEs depend on this choice and on the fact that the reference is the operator network output, not a true FEM solution.
assumptions (6)
  • standard math Unique continuation for elliptic equations: a solution vanishing on an open subdomain must vanish on the connected domain.
    Used in Section 3.1 to prove uniqueness of the L2(omega) projection and the coefficient stability estimate (3.8).
  • domain assumption Boundary data of the target solution belong to a finite-dimensional set G or its delta-neighborhood, and the POD/autoencoder trained on GS recovers this set.
    Introduced in Eq. (2.3) and used in Section 4.1. It is the mechanism that converts logarithmic ill-posedness into Lipschitz stability; no error bound links training representability to inverse reconstruction error.
  • standard math Elliptic regularity estimates: ||phi_n||_{H2(Omega)} is bounded by C ||g_n||_{H3/2(dOmega)} and H2 regularity for the dual problem hold on the domain.
    Invoked for the Nitsche approximation (3.12) and in the duality argument of Lemma 3.1 in Sections 3.2 and 3.3. Fails on non-smooth domains, which would break the h-scaling.
  • domain assumption The nonlinear energy functional E(v)=1/2 integral (1+v^2)|grad v|^2 defines a well-posed FE problem for every boundary coefficient, and the energy loss is trainable.
    The operator network loss in Eq. (4.9) is built on this energy; the paper does not prove existence, uniqueness, or convexity in the FE space for the nonlinear operator.
  • ad hoc to paper After training, the autoencoder decoder and operator network approximate the boundary manifold and the FE solution map closely enough for latent optimization to recover the true solution.
    The nonlinear numerical claim depends on this unproved approximation property. Table 2 gives empirical network versus FEM errors, but no bound connects them to inverse reconstruction, and the nonlinear reference solutions are network outputs.
  • domain assumption The sampled collective data GS is representative of the target realization's boundary data.
    POD and autoencoder are trained only on GS; if the target lies outside the learned manifold, the latent optimization (4.12) cannot represent it. This is a data coverage assumption, not proved.
invented entities (1)
  • Latent space Z (autoencoder bottleneck)
    purpose: Low-dimensional parametrization of boundary data coefficients; the inverse problem is optimized over this space instead of the coefficient space.
    A computational construct, not a physical entity. Its usefulness is shown only by reconstruction error on training-like data; there is no independent falsifiable prediction, so it carries the burden of data representability.

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Cite this review

Pith. "Pith review of Stabilizing and Solving Unique Continuation Problems by Parameterizing Data and Learning Finite Element Solution Operators." pith.science (2026). https://pith.science/paper/TWND27QP

@misc{pith2026241204409,
  author       = {Pith},
  title        = {Pith review of: Stabilizing and Solving Unique Continuation Problems by Parameterizing Data and Learning Finite Element Solution Operators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TWND27QP}},
  note         = {Machine review of arXiv:2412.04409}
}
abstract

We consider an inverse problem involving the reconstruction of the solution to a nonlinear partial differential equation (PDE) with unknown boundary conditions. Instead of direct boundary data, we are provided with a large dataset of boundary observations for typical solutions (collective data) and a bulk measurement of a specific realization. To leverage this collective data, we first compress the boundary data using proper orthogonal decomposition (POD) in a linear expansion. Next, we identify a possible nonlinear low-dimensional structure in the expansion coefficients using an autoencoder, which provides a parametrization of the dataset in a lower-dimensional latent space. We then train an operator network to map the expansion coefficients representing the boundary data to the finite element (FE) solution of the PDE. Finally, we connect the autoencoder's decoder to the operator network which enables us to solve the inverse problem by optimizing a data-fitting term over the latent space. We analyze the underlying stabilized finite element method (FEM) in the linear setting and establish an optimal error estimate in the $H^1$-norm. The nonlinear problem is then studied numerically, demonstrating the effectiveness of our approach.

Figures

Figures reproduced from arXiv: 2412.04409 by the authors.

Figure 1
Figure 1. Schematic view of the minimization problem setup where we seek the v ∈ V that minimizes the error in the observation of the solution, while under a PDE constraint with boundary conditions according to experience. 2.2 Finite Element Method Let Vh be a finite element space on a quasi-uniform partition Th of Ω into shape regular elements with mesh parameter h ∈ (0, h0] and assume that there is an interpolation operator… view at source ↗
Figure 2
Figure 2. Illustrations for the analytical example with exponential growth of the stability constant. In (a), we show the unit disc domain containing a subdomain ω, in the form of a centered disc of radius rω. Looking at the first five non-zero modes φ2n+1 = r n sin(nθ) from the expansion (3.60) we see that these modes rapidly becomes very small within ω, making the problem of retrieving the coefficient values in the expansio… view at source ↗
Figure 3
Figure 3. Overview of networks utilized in methods based on machine learning. The autoencoder network in (a) is used for identifying a low￾dimensional structure in the dataset G. The operator network in (b) is trained to approximate the solution to the PDE, given input boundary data. The composition of the decoder part of the autoencoder and the operator network in (c) is used for solving the inverse problem in the low-dimens… view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: Domain used in all numerical examples with the subdomain ω indicated. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]
Figure 5
Figure 5. Figure 5: FEM interior basis functions with their corresponding POD boundary basis functions on a structured uniform 82x82 triangular mesh of the unit square. This enables us to solve a linear inverse minimization problem over the coefficients (c1, ..., cN ) ∈ R N in the linear …
Figure 6
Figure 6. Figure 6: Optimization process over a 9-dimensional coefficient space for a linear inverse problem with noisy data. Here, the FEM interior basis functions in [PITH_FULL_IMAGE:figures/full_fig_p020_6.png]
Figure 7
Figure 7. Figure 7: Optimization process over a 21-dimensional coefficient space for a nonlinear inverse problem with noisy data. Here, the operator network in the last row of Table 2b (21 input coefficients, 59049 output DoFs) is used. The unperturbed data is shown in the first frame. Th…
Figure 8
Figure 8. Figure 8: Coefficients versus parameters for the linear case (left) and the quadratic case (right). All parameters have the same value which varies be￾tween −2 and 2. For each case, there is an upper and lower bound for the coefficients obtained by taking all matrix entries to e…
Figure 9
Figure 9. Figure 9: Left: PCA results for both the linear and quadratic case. The plots show the singular values of the coefficients in decreasing order for three different perturbations: unperturbed and two perturbed (standard deviation = 0.5 and 1). Right: Autoencoder results for the qu…
Figure 10
Figure 10. Figure 10: Gaussian data examples. Left: Bell curves used for the coef￾ficients, unperturbed (blue) and perturbed (red). Middle: PCA results for unperturbed data (blue) and perturbed (red, standard deviation = 0.15). The plots show the singular values of the coefficients in decr…
Figure 11
Figure 11. Figure 11: Optimization process over a 9-dimensional latent space for a nonlinear inverse problem with clean data. Again, the operator network in the last row of Table 2b (21 input coefficients, 59049 output DoFs) is used, but here together with the “sd = 0” decoder from the rig…
Figure 12
Figure 12. Figure 12: Optimization process over a 9-dimensional latent space for a nonlinear inverse problem with noisy data. Again, the operator network in the last row of Table 2b (21 input coefficients, 59049 output DoFs) is used together with the “sd = 0” decoder from the right frame i…
Figure 13
Figure 13. Figure 13: Optimization process over a 9-dimensional latent space for a nonlinear inverse problem with noisy data. Again, the operator network in the last row of Table 2b (21 input coefficients, 59049 output DoFs) is used, but here together with the “sd = 0.15” decoder from the …

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Works this paper leans on

45 extracted references · 18 canonical work pages

  1. [1]

    Alessandrini, L

    G. Alessandrini, L. Rondi, E. Rosset, and S. Vessella. The stability for the Cauchy problem for elliptic equations. Inverse Probl., 25(12):123004, 47, 2009. doi:10.1088/0266-5611/25/12/123004

    work page doi:10.1088/0266-5611/25/12/123004 2009

  2. [2]

    Arridge, P

    S. Arridge, P. Maass, O. ¨Oktem, and C.-B. Sch¨ onlieb. Solving inverse problems using data-driven models. Acta Numer., 28:1–174, 2019. doi:10.1017/S0962492919000059

    doi:10.1017/s0962492919000059 2019

  3. [3]

    Bachmayr, W

    M. Bachmayr, W. Dahmen, and M. Oster. Variationally correct neural residual regres- sion for parametric PDEs: On the viability of controlled accuracy. arXiv Preprint , 2024. doi:10.48550/arXiv.2405.20065

    arXiv doi:10.48550/arxiv.2405.20065 2024

  4. [4]

    Berg and K

    J. Berg and K. Nystr¨ om. Neural networks as smooth priors for inverse problems for PDEs. J. Comput. Appl. Math. Data Sci. , 1:100008, 2021. doi:10.1016/j.jcmds.2021.100008

    arXiv 2021

  5. [5]

    Boulakia, C

    M. Boulakia, C. James, and D. Lombardi. Numerical approximation of the unique continuation problem enriched by a database for the Stokes equations. Preprint, Oct. 2024. https://inria. hal.science/hal-04721560

    work page 2024

  6. [6]

    Bourgeois

    L. Bourgeois. A mixed formulation of quasi-reversibility to solve the Cauchy problem for Laplace’s equation. Inverse Probl., 21(3):1087–1104, 2005. doi:10.1088/0266-5611/21/3/018

    work page doi:10.1088/0266-5611/21/3/018 2005

  7. [7]

    E. Burman. Stabilized finite element methods for nonsymmetric, noncoercive, and ill-posed problems. Part I: Elliptic equations. SIAM J. Sci. Comput. , 35(6):A2752–A2780, 2013. doi:10.1137/130916862. 32

    work page doi:10.1137/130916862 2013

  8. [8]

    E. Burman. Error estimates for stabilized finite element methods applied to ill-posed problems. C. R. Math. Acad. Sci. Paris , 352(7-8):655–659, 2014. doi:10.1016/j.crma.2014.06.008

    work page doi:10.1016/j.crma.2014.06.008 2014

Show all 45 references
  1. [9]

    Burman, P

    E. Burman, P. Hansbo, and M. G. Larson. Solving ill-posed control problems by stabilized finite element methods: an alternative to Tikhonov regularization. Inverse Probl., 34(3):035004, 36, 2018. doi:10.1088/1361-6420/aaa32b

    2018 doi

  2. [10]

    Burman, P

    E. Burman, P. Hansbo, M. G. Larson, and K. Larsson. Isogeometric analysis and augmented La- grangian Galerkin least squares methods for residual minimization in dual norm. Comput. Methods Appl. Mech. Engrg. , 417(part B):Paper No. 116302, 17, 2023. doi:10.1016/j.cma.2023.116302

    2023

  3. [11]

    Burman, M

    E. Burman, M. Nechita, and L. Oksanen. Optimal approximation of unique continuation. Found. Comput. Math. , 2024. doi:10.1007/s10208-024-09655-w

    2024 doi

  4. [12]

    Burman and L

    E. Burman and L. Oksanen. Finite element approximation of unique continuation of func- tions with finite dimensional trace. Math. Models Methods Appl. Sci. , 34(10):1809–1824, 2024. doi:10.1142/S0218202524500362

    2024 doi

  5. [13]

    Burman, L

    E. Burman, L. Oksanen, and Z. Zhao. Computational unique continuation with finite dimensional Neumann trace. arXiv Preprint, 2024. doi:10.48550/arXiv.2402.13695

    2024 doi arXiv

  6. [14]

    S. Cen, B. Jin, Q. Quan, and Z. Zhou. Hybrid neural-network fem approximation of diffu- sion coefficient in elliptic and parabolic problems. IMA J. Numer. Anal. , 44(5):3059–3093, 2023. doi:10.1093/imanum/drad073

    2023 doi

  7. [15]

    S. Cen, B. Jin, K. Shin, and Z. Zhou. Electrical impedance tomography with deep Calder´ on method. J. Comput. Phys. , 493:112427, 2023. doi:10.1016/j.jcp.2023.112427

    2023

  8. [16]

    Chung, K

    E. Chung, K. Ito, and M. Yamamoto. Least squares formulation for ill-posed inverse problems and applications. Appl. Anal., 101(15):5247–5261, 2022. doi:10.1080/00036811.2021.1884228

    2022

  9. [17]

    T. Cui, Y. M. Marzouk, and K. E. Willcox. Data-driven model reduction for the Bayesian solution of inverse problems. Int. J. Numer. Methods Eng. , 102(5):966–990, 2015. doi:10.1002/nme.4748

    2015 doi

  10. [18]

    Dahmen, H

    W. Dahmen, H. Monsuur, and R. Stevenson. Least squares solvers for ill-posed PDEs that are conditionally stable. ESAIM Math. Model. Numer. Anal. , 57(4):2227–2255, 2023. doi:10.1051/m2an/2023050

    2023

  11. [19]

    Dasgupta, D

    A. Dasgupta, D. V. Patel, D. Ray, E. A. Johnson, and A. A. Oberai. A dimension-reduced variational approach for solving physics-based inverse problems using generative adversarial net- work priors and normalizing flows. Comput. Methods Appl. Mech. Eng. , 420:116682, 2024. doi:...

    2024

  12. [20]

    M. V. de Hoop, D. Z. Huang, E. Qian, and A. M. Stuart. The cost-accuracy trade-off in operator learning with neural networks. J. Mach. Learn., 1(3):299–341, 2022. doi:10.48550/arXiv.2203.13181

    2022 doi arXiv

  13. [21]

    N. Demo, M. Strazzullo, and G. Rozza. An extended physics informed neural network for prelim- inary analysis of parametric optimal control problems. Comput. Math. Appl. , 143:383–396, 2023. doi:10.1016/j.camwa.2023.05.004

    2023 doi

  14. [22]

    Escapil-Inchausp´ e and G

    P. Escapil-Inchausp´ e and G. A. Ruz. h-analysis and data-parallel physics-informed neural networks. Sci. Rep., 13(17562), 2023. doi:10.1038/s41598-023-44541-5

    2023 doi

  15. [23]

    N. R. Franco, A. Manzoni, and P. Zunino. Mesh-informed neural networks for operator learning in finite element spaces. J. Sci. Comput. , 97(2):Paper No. 35, 41, 2023. doi:10.1007/s10915-023- 02331-1

    2023 doi

  16. [24]

    Fulton, V

    L. Fulton, V. Modi, D. Duvenaud, D. I. W. Levin, and A. Jacobson. Latent-space Dynam- ics for Reduced Deformable Simulation. Comput. Graph. Forum , 38(2):379–391, May 2019. doi:10.1111/cgf.13645

    2019 doi

  17. [25]

    Z. Gao, L. Yan, and T. Zhou. Adaptive operator learning for infinite-dimensional Bayesian inverse problems. SIAM/ASA J. Uncertain. Quantif. , 12(4):1389–1423, 2024. doi:10.1137/24M1643815

    2024 doi

  18. [26]

    Ghattas and K

    O. Ghattas and K. Willcox. Learning physics-based models from data: perspectives from inverse problems and model reduction. Acta Numer., 30:445–554, 2021. doi:10.1017/S0962492921000064. 33

    2021 doi

  19. [27]

    Ivagnes, N

    A. Ivagnes, N. Demo, and G. Rozza. Towards a machine learning pipeline in reduced order modelling for inverse problems: neural networks for boundary parametrization, dimensionality reduction and solution manifold approximation. J. Sci. Comput., 95(1):Paper No. 23, 24, 2023. do...

    2023 doi

  20. [28]

    C. James. L’interaction entre donn´ ees et mod´ elisation pour le probl` eme d’assimilation de donn´ ees. Master’s thesis, Sorbonne Universit´ e, 2023. Rapport de stage de Master 2

    2023

  21. [29]

    Kovachki et al

    N. Kovachki et al. Neural operator: learning maps between function spaces with applications to PDEs. J. Mach. Learn. Res. , 24:Paper No. [89], 97, 2023. doi:10.48550/arXiv.2108.08481

    2023 doi arXiv

  22. [30]

    M. G. Larson, C. Lundholm, and A. Persson. Nonlinear operator learning using energy minimization and MLPs. arXiv Preprint, 2024. doi:10.48550/arXiv.2412.04596

    2024 doi arXiv

  23. [31]

    Lieberman, K

    C. Lieberman, K. Willcox, and O. Ghattas. Parameter and state model reduction for large-scale statistical inverse problems. SIAM J. Sci. Comput. , 32(5):2523–2542, 2010. doi:10.1137/090775622

    2010 doi

  24. [32]

    Logg, K.-A

    A. Logg, K.-A. Mardal, G. N. Wells, et al. Automated Solution of Differential Equations by the Finite Element Method . Springer, 2012. doi:10.1007/978-3-642-23099-8

    2012 doi

  25. [33]

    S. Lunz, A. Hauptmann, T. Tarvainen, C.-B. Sch¨ onlieb, and S. Arridge. On learned operator correction in inverse problems. SIAM J. Imaging Sci., 14(1):92–127, 2021. doi:10.1137/20M1338460

    2021 doi

  26. [34]

    S. A. McQuarrie, P. Khodabakhshi, and K. E. Willcox. Nonintrusive reduced-order models for parametric partial differential equations via data-driven operator inference. SIAM J. Sci. Comput. , 45(4):A1917–A1946, 2023. doi:10.1137/21M1452810

    2023 doi

  27. [35]

    Mishra and R

    S. Mishra and R. Molinaro. Estimates on the generalization error of physics-informed neural net- works for approximating a class of inverse problems for PDEs. IMA J. Numer. Anal. , 42(2):981– 1022, 2022. doi:10.1093/imanum/drab032

    2022 doi

  28. [36]

    M. Nechita. Solving ill-posed Helmholtz problems with physics-informed neural networks. J. Numer. Anal. Approx. Theory, 52(1):90–101, 2023. doi:10.33993/jnaat521-1305

    2023 doi

  29. [37]

    Pakravan, P

    S. Pakravan, P. A. Mistani, M. A. Aragon-Calvo, and F. Gibou. Solving inverse-PDE prob- lems with physics-aware neural networks. J. Comput. Phys. , 440:Paper No. 110414, 31, 2021. doi:10.1016/j.jcp.2021.110414

    2021

  30. [38]

    Patel, D

    D. Patel, D. Ray, M. R. Abdelmalik, T. J. Hughes, and A. A. Oberai. Variationally mimetic operator networks. Comput. Methods Appl. Mech. Eng. , 419:116536, 2024. doi:10.1016/j.cma.2023.116536

    2024

  31. [39]

    Raissi, P

    M. Raissi, P. Perdikaris, and G. E. Karniadakis. Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J. Comput. Phys. , 378:686–707, 2019. doi:10.1016/j.jcp.2018.10.045

    2019 doi

  32. [40]

    D. Ray, O. Pinti, and A. A. Oberai. Deep Learning and Computational Physics . Springer Cham, Switzerland, 2024. doi:10.1007/978-3-031-59345-1

    2024 doi

  33. [41]

    Riffaud, M

    S. Riffaud, M. A. Fern´ andez, and D. Lombardi. A low-rank solver for parameter estimation and un- certainty quantification in time-dependent systems of partial differential equations. J. Sci. Comput., 99(2):Paper No. 34, 31, 2024. doi:10.1007/s10915-024-02488-3

    2024 doi

  34. [42]

    Romor, G

    F. Romor, G. Stabile, and G. Rozza. Non-linear manifold reduced-order models with convolutional autoencoders and reduced over-collocation method. J. Sci. Comput. , 94(3):Paper No. 74, 39, 2023. doi:10.1007/s10915-023-02128-2

    2023 doi

  35. [43]

    Sharp et al

    N. Sharp et al. Data-free learning of reduced-order kinematics. In Special Interest Group on Computer Graphics and Interactive Techniques Conference Conference Proceedings, pages 1–9, Los Angeles CA USA, July 2023. ACM. doi:10.1145/3588432.3591521

    2023

  36. [44]

    T. Xu, D. Liu, P. Hao, and B. Wang. Variational operator learning: a unified paradigm marrying training neural operators and solving partial differential equations. J. Mech. Phys. Solids, 190:Paper No. 105714, 29, 2024. doi:10.1016/j.jmps.2024.105714

    2024

  37. [45]

    Zhang, T

    L. Zhang, T. Luo, Y. Zhang, W. E., Z.-Q. J. Xu, and Z. Ma. MOD-Net: a machine learning approach via model-operator-data network for solving PDEs. Commun. Comput. Phys. , 32(2):299–335, 2022. doi:10.4208/cicp.oa-2021-0257. 34 Authors’ addresses: Erik Burman, Mathematics, Univer...

    2022 doi

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.