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Physics-Informed Deep Inverse Operator Networks for Solving PDE Inverse Problems

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

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Pith's one-line read PI-DIONs learn the map from partial measurements of a PDE solution to the unknown field—the inverse operator—with no labeled training data, and the paper proves that small physics-informed loss guarantees small prediction error over the…

desk verdict Nice architecture and promising empirical results, but the advertised stability-based generalization guarantee doesn't survive contact with the actual loss: the norm gap is real, and Theorem 3's probability bound is malformed. read the letter →

arxiv 2412.03161 v2 pith:CFXVORTA submitted 2024-12-04 math.NA cs.AIcs.NA

Sung Woong Cho , Hwijae Son This is my paper · ORCID
classification math.NAcs.AIcs.NA MSC 35R3065M3268T07
keywords inverseproblemsoperatorlearningphysics-informedneuralnetworksstabilityestimatespartialdifferentialequationsunsupervisedDeepONetgeneralizationbounds
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 proposes PI-DIONs, a network architecture that learns the solution operator of PDE-based inverse problems—the map from partial measurements of a solution to the unknown field—without any labeled training data. The training loss combines a physics residual enforcing the PDE and boundary conditions with a data term matching the measurements, so no paired examples are needed. The paper's central theoretical claim is that stability estimates from the inverse-problem literature extend to this operator-learning setting: with enough training samples and grid points, small empirical loss implies small prediction error over the whole domain with high probability. The authors argue this makes real-time inference from sparse measurements reliable, and support it with experiments on reaction-diffusion, Helmholtz, and Darcy-flow inverse problems.

What carries the argument

The load-bearing object is the stability estimate linking physically meaningful residuals to $L^2$ errors: for a solution $u$ and unknown $s$, the total prediction error is bounded by the PDE residual, boundary residual, and measurement mismatch, plus a discretization term that vanishes as the grids refine. The paper extends this one-instance inequality to the operator setting by proving (Theorems 1 and 2) that the empirical loss concentrates around its continuous expectation using a symmetrization argument with Hoeffding's inequality and covering numbers of the network-realized function classes; Theorem 3 then composes this concentration with the stability estimate. The architecture itself is a physics-informed DeepONet variant: a trunk network produces basis functions of the spatial/temporal coordinate, a reconstruction branch turns partial measurements into coefficients for $u$, and an inverse branch turns the same measurements into coefficients for $s$, so automatic differentiation computes the PDE residual at collocation points.

What would settle it

Train PI-DION on an inverse problem without a valid single-instance stability estimate (for example, a non-unique source recovery), drive both loss terms below the theorem's thresholds, and check whether test prediction errors stay below $\sqrt{\varepsilon}$ at the claimed frequency; if they do not, the stability-based generalization claim is specific to problems with such estimates, not general. A cheaper check is to estimate the covering number of the trained networks and verify whether the sample counts used in the experiments actually meet the theorem's requirements.

Watch

Extended reading notes

Core claim

PI-DIONs parameterize the reconstructed solution and the unknown target as trunk-network basis expansions, with two branch networks reading the same partial measurement, so the physics loss $L_{\text{physics}}$ acts on the functions themselves rather than on paired input-output labels. The main theoretical result, Theorem 3, states that for any $u\in U$ and $s\in S$, the prediction error satisfies $\lVert u_{\eta,\theta}-u\rVert_{L^2(\Omega)} + \lVert s_{\zeta,\theta}-s\rVert_{L^2(\Omega)} \le \sqrt{\varepsilon}$ with probability at least $(1-2\delta)(1-2\sqrt{\varepsilon}-\widetilde{L}_{\text{physics}}-\widetilde{L}_{\text{data}})/\sqrt{\varepsilon}$, provided the numbers of sampled functions and grid points meet the thresholds in Theorems 1 and 2 and the network-realized function classes have finite covering numbers. This extends single-instance stability estimates for reaction-diffusion (Zhang et al. 2023) and Helmholtz (Zhang & Liu 2023a) inverse problems to a probabilistic statement over entire function spaces. The accompanying universal approximation result (Proposition 1) asserts the physics-informed loss can be driven arbitrarily small, and the reported experiments show unsupervised PI-DION matching or beating supervised DeepONet and FNO baselines on all three benchmarks.

Load-bearing premise

The proof assumes the class of functions representable by the branch and trunk networks has a finite covering number, and that the trunk network can approximate true solutions and their derivatives to arbitrary accuracy; no bound or construction is supplied, and if the realized class is too complex the claimed generalization guarantee does not follow.

Editorial extensions

If this is right

  • Partial measurements alone—with no labeled pairs—are enough to train an inverse operator that predicts the unknown field at arbitrary resolution, removing the main data bottleneck of supervised operator learning.
  • Theorem 3 gives an explicit, if not numerically sharp, prescription for how many training functions and collocation points are needed to certify a target accuracy with a target confidence.
  • For any inverse problem admitting a single-instance stability estimate, the paper's extension yields the same probabilistic guarantee, so the method is not tied to the three benchmark equations tested.
  • The supervised variant of PI-DION outperforms supervised DeepONet and FNO baselines in the reported experiments, while the unsupervised variant matches or nearly matches them, so labeled data can be used when available but is not required.
  • A variable-input extension handles irregular sensor locations, which is the common practical setting for measurement data.

Reading between the lines

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

  • Inference: The same proof recipe should transfer to any inverse problem with a conditional stability estimate, making the architecture secondary to the availability of such an estimate; for severely ill-posed problems without stability, the claims in Theorem 3 have no domain of application.
  • Inference: The covering number $N_c$ in Theorems 1 and 2 is never quantified, so the certified sample sizes could be impractically large for realistic networks; empirically checking whether the 1000-sample experiments operate within the theorem's regime would clarify how tight the theory is.
  • Inference: Because both branch networks share the same trunk and measurement input, the architecture naturally supports uncertainty quantification by replacing the deterministic coefficient outputs with distributions, which would enable credible intervals on the reconstructed field without changing the physics loss.
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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

5 major / 5 minor

Summary. The paper introduces Physics-Informed Deep Inverse Operator Networks (PI-DIONs), an unsupervised operator-learning architecture for PDE-based inverse problems. The model consists of a reconstruction branch, an inverse branch, and a trunk network; the branches ingest partial measurements and output coefficients that, combined with the trunk, produce both the reconstructed solution u_{η,θ} and the unknown target s_{ζ,θ}. Training minimizes a physics residual loss plus a measurement-data loss, with no labeled target pairs. The authors claim that stability estimates from the inverse-problems literature can be extended to the operator-learning setting, yielding finite-sample and finite-grid generalization guarantees (Theorems 1-3), and they provide a universal approximation result (Proposition 1). Experiments on reaction-diffusion, Helmholtz, and Darcy-flow inverse problems compare unsupervised and supervised PI-DIONs against supervised DeepONet and FNO baselines.

Significance. If the theoretical claims were correct, the paper would make a valuable step toward unsupervised operator learning for inverse problems, with real-time inference and no labeled data. The empirical results are suggestive and the reproducibility statement indicates that source code is provided. However, the central theoretical foundation is not established: the main generalization theorem is not a well-formed probability statement, the stability estimates are not connected to the actual training loss, and the proofs rely on unproved or circular assumptions. These are load-bearing issues for the paper's headline contribution, not presentation defects. The empirical comparison is also limited by the absence of unsupervised baselines and error bars.

major comments (5)
  1. [Section 3.2, Theorem 3 and Appendix D.3] Theorem 3 is not a valid probability statement. The expression '(1 − 2δ)(1 − 2√ϵ − Lphysics+Ldata)√ϵ )' contains a stray parenthesis, treats the random empirical losses Lphysics and Ldata as if they were constants inside the probability expression, and does not specify the probability space over which 'for any u ∈ U, s ∈ S' is evaluated. More substantively, Appendix D.3 uses Markov's inequality to bound the measure of inputs whose continuous residual is below α, then concludes a bound on prediction error for 'u ∈ U' by invoking a stability estimate for a single element. This does not prove the claimed high-probability, uniform-over-U error bound; the stability estimate is applied to an arbitrary element whose residual is small, but the theorem as stated quantifies over all u and s simultaneously. The proof never bridges the norm gap described in the next comment, so the advertised finite-sample/finite-grid generalization guarantee is unsupported even in the infinite-sample, infinite-grid limit.
  2. [Section 3.1 versus Section 2 and Appendix D.3] There is a norm mismatch between the stability estimates and the training loss. For the reaction-diffusion problem, the stability bound in Section 3.1 controls ∥f∗−f∥_{L2(Ω)} + ∥u∗−u∥_{C([0,T];L2(Ω))} by a residual measured in H^1(0,T;L2(Ω)), together with Laplacian terms at initial and final times and an H^2 boundary term. The training loss Lphysics is an L^2 (squared) interior residual plus an L^2 boundary residual, and Ldata is an L^2 measurement residual; no H^1 residual, Laplacian term, or H^2 boundary term appears. Smallness of an L^2 norm does not imply smallness of an H^1 norm, so the right-hand side of the stability estimate is not controlled by the trained loss. The same issue affects the Helmholtz stability bound, which uses Hölder powers of residuals and an extra term on Ω\(Ω_m ∪ Ω_0) that is absent from the loss. Appendix D.3 does not address this gap: it simply states 'Recalling stability estimates for a single element' and immediately bounds the L^2 error by α. This missing link is central to the paper's claim that Theorem 3 guarantees generalization.
  3. [Appendix D.1, Theorems 1 and 2] The generalization bounds for Ldata and Lphysics depend on covering numbers Nc, NN, and NB that are assumed finite in Definition 1 and Definition 2, but no bound, construction, or example is provided. If the realized function classes have infinite covering numbers, the probability bounds in Propositions 2-4 are vacuous. In addition, the symmetrization proof has internal problems: in Lemma 1 the text states that 'the event on the left-hand side includes the event on the right-hand side,' which is the reverse of what symmetrization requires, and the displayed chain in the proof of Lemma 2 appears to conflate the two halves of the permuted sample. These are not merely cosmetic issues, because Theorems 1 and 2 are the finite-sample foundation on which Theorem 3 rests.
  4. [Section 3.3, Proposition 1 and Appendix D.4] The universal approximation argument is circular or incomplete. Proposition 1 claims that for any dataset and any ε>0 there exist parameters such that L = Lphysics + Ldata ≤ ε. The proof in Appendix D.4 first approximates the operator G on a finite dataset, which only controls Ldata, and then separately constructs branch outputs as one-hot encodings and a trunk that approximates (u_1,...,u_N,s_1,...,s_N) with derivatives. Standard multi-layer perceptron universal approximation theorems do not, without additional hypotheses, guarantee simultaneous approximation of derivatives; no such Sobolev-type approximation theorem is cited or proved. For nonlinear differential operators N involving products, the argument that boundedness of u suffices to control products of approximation errors is not developed, so the claimed reduction of Lphysics is not established. Since Proposition 1 is needed to justify that the small-loss regime required by Theorem 3 is reachable, this is a load-bearing gap.
  5. [Section 4, Table 1] The empirical evaluation compares PI-DIONs only against supervised DeepONet and FNO baselines. There is no unsupervised baseline such as PI-DeepONet or PINO, so the paper's central practical claim—that PI-DIONs learn inverse operators 'without labeled data'—is not benchmarked against existing unsupervised methods. In addition, Table 1 reports single relative L2 error numbers without standard deviations over multiple seeds or dataset draws, which makes it difficult to assess whether the reported differences, especially the 1.03% versus 1.04% reaction-diffusion results, are meaningful.
minor comments (5)
  1. [Section 3.1] The reaction-diffusion stability estimate contains an unmatched parenthesis in '∥∆(u(x, T) − u∗(x, T)∥'; the closing parenthesis after 'u∗(x, T)' is missing. The displayed inequality is also missing a closing parenthesis before '≤ CR'.
  2. [Section 3.2, Theorem 1] The sample-complexity condition for N is written as N ≥ 8 log(8Nc/δ)/log 2, but Nc is defined as a constant depending on ε; the dependence of the right-hand side on ε should be stated explicitly. The proof in Appendix D.1 yields a bound with N·L in the exponent (Proposition 2), which is a different statement from the theorem's displayed condition, so the two should be reconciled.
  3. [Section 3.2] The phrase 'for arbitrary u, s sampled from µ(U) and ν(S)' is internally contradictory: 'arbitrary' elements are not 'sampled' elements. The intended quantifier structure of Theorems 1-3 should be clarified consistently.
  4. [Appendix C.2, Table 6] The sensitivity analysis reports relative L2 errors for seven weight combinations but does not state the number of random seeds used; given the small differences in some entries, a seed-average with a standard deviation would be more informative.
  5. [Section 6] The reproducibility statement says source code is submitted as supplementary material, but the paper text does not include a link or a persistent repository identifier. For a journal version, a DOI or stable URL should be provided.

Circularity Check

2 steps flagged · score 6.0 of 10

Theorem 3's generalization guarantee is obtained by invoking an informal single-element stability estimate that is the theorem's conclusion itself; Proposition 1 likewise assumes the derivative-approximation capability it is meant to prove.

  1. self definitional [Appendix D.3, proof of Theorem 3; compare the formal stability estimates in Section 3.1]
    "Recalling stability estimates for a single element inU, we can conclude that, with probability at least 1 − ( ˜Lphysics + ˜Ldata)/α, the following error bound holds for u ∈ U. ∥uη,θ − u∥L2(Ω) + ∥sζ,θ − s∥L2(Ω) ≤ α."

    The event proved by Markov is that the continuous L^2 residuals are at most α; to conclude the displayed error bound, the proof needs a stability estimate of exactly the form error ≤ residual. The formal single-element estimates imported in Section 3.1 do not have that form: the reaction–diffusion estimate bounds error by an H^1(0,T;L^2) residual norm plus Laplacian initial/final-data terms and an H^2 boundary term, and the Helmholtz estimate uses Hölder powers C2|Ωm|/|Ω| of residuals. None of these terms is controlled by L̃physics + L̃data. The only estimate in the paper with the needed form is the informal 'stability estimate' stated in Section 3.1, which is precisely the single-element version of Theorem 3's conclusion.

  2. other [Appendix D.4, proof of Proposition 1]
    "By the universal approximation theorem for neural networks, for any multi-index α, we have: X_{|α|<=n} ∂^α_x |τ (x) − (u_1(x), s_1(x), · · ·, u_N (x), c_N (x))| ≤ ϵ."

    Proposition 1 asserts existence of parameters with L = Lphysics + Ldata ≤ ε. The proof constructs the branch networks as coordinate selectors and then asserts, by the universal approximation theorem, that the trunk network τ approximates (u_1,...,u_N, s_1,...,s_N) together with all derivatives up to order n. That joint function-plus-derivative approximation is precisely the capability needed to make the physics residual small; it is the substantive content of the proposition, and it is not supplied by the cited Chen & Chen operator universal approximation theorem, which concerns function values only. The proposition is therefore established by assuming the very approximation property it was meant to guarantee.

full rationale

The circularity is concentrated in the theoretical generalization claims. Theorem 3's proof in Appendix D.3 first uses Markov's inequality to obtain an event on which the continuous squared residuals are small, then 'recalls stability estimates for a single element' and jumps directly to the desired L^2 error bound. The only single-element stability estimate in the paper with that exact error-controls-residual form is the informal statement in Section 3.1, which is the conclusion of the theorem in the single-element case; the formal estimates from Zhang et al. (2023) and Zhang & Liu (2023a) involve H^1 residuals, Laplacians of initial/final data, H^2 boundary norms, and Hölder powers that are not bounded by the L^2 losses actually minimized. So the advertised finite-sample generalization guarantee reduces, in the proof, to an assumption equivalent to the theorem's claim. Proposition 1 has a parallel issue: its proof assumes that a trunk network can approximate all target solutions and sources together with all derivatives up to order n, which is the very approximation capability that must be proved in order to show Lphysics can be made small. These are not cases of harmless self-citation: the self-citations in the paper (e.g., Cho et al. 2024) appear only as future-work remarks and are not load-bearing, and the stability estimates are cited from non-overlapping authors. The empirical comparisons against supervised baselines are independent and not circular. Correctness problems noted by the skeptic pass, such as the norm gap between L^2 losses and H^1 stability bounds and the malformed probability expression in Theorem 3, are related symptoms but are separate from circularity. Because the circular step sits at the center of the paper's main theoretical contribution, the score is 6 rather than a lower value.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

The central theoretical claim rests on imported stability estimates and unverified complexity assumptions about the network function classes. The only hand-tuned parameters are the loss weights, which are not derived. No new physical entities are introduced.

free parameters (1)
  • Loss weights λ1, λ2 = λ1=1, λ2=100
    Chosen by hand based on sensitivity analysis (Table 6) and the remark that larger λ2/λ1 gives smaller test error; effectively tuned on the test set, not derived from theory.
assumptions (3)
  • domain assumption Stability estimates for reaction-diffusion and Helmholtz inverse source problems (from Zhang et al. 2023 and Zhang & Liu 2023a) hold with the stated constants.
    Invoked in Section 3.1 and used directly in Theorem 3; the paper does not re-derive them.
  • ad hoc to paper The function classes {uη,θ} and {sζ,θ} have finite covering numbers Nc, NN, NB.
    Assumed in Theorem 1 and 2 proofs (Appendix D.1, Definition 1, where Nc is allowed to be infinite but the theorem requires finite); no bound is proven for the specific networks.
  • domain assumption The solution and inverse operators are stable/Lipschitz with respect to the measurement, so that the stability estimates can be applied after training.
    Used in Proposition 1 and the conclusion of Theorem 3; implicitly assumes the inverse problem is well-posed in the sense of the stability estimate.

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Cite this review

Pith. "Pith review of Physics-Informed Deep Inverse Operator Networks for Solving PDE Inverse Problems." pith.science (2026). https://pith.science/paper/CFXVORTA

@misc{pith2026241203161,
  author       = {Pith},
  title        = {Pith review of: Physics-Informed Deep Inverse Operator Networks for Solving PDE Inverse Problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CFXVORTA}},
  note         = {Machine review of arXiv:2412.03161}
}
read the original abstract

Inverse problems involving partial differential equations (PDEs) can be seen as discovering a mapping from measurement data to unknown quantities, often framed within an operator learning approach. However, existing methods typically rely on large amounts of labeled training data, which is impractical for most real-world applications. Moreover, these supervised models may fail to capture the underlying physical principles accurately. To address these limitations, we propose a novel architecture called Physics-Informed Deep Inverse Operator Networks (PI-DIONs), which can learn the solution operator of PDE-based inverse problems without labeled training data. We extend the stability estimates established in the inverse problem literature to the operator learning framework, thereby providing a robust theoretical foundation for our method. These estimates guarantee that the proposed model, trained on a finite sample and grid, generalizes effectively across the entire domain and function space. Extensive experiments are conducted to demonstrate that PI-DIONs can effectively and accurately learn the solution operators of the inverse problems without the need for labeled data.

Figures

Figures reproduced from arXiv: 2412.03161 by the authors.

Figure 1
Figure 1. Schematic illustration of PI-DIONs architecture. The reconstruction and inverse branch [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. A test sample and results for the inverse source problem of the reaction-diffusion equation. [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. A test sample and results for the Helmholtz equation. (a) Partial measurement [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: A test sample and results for the Darcy flow. (a) Full measurement [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: Schematic illustration of PI-DIONs-v0 architecture. The inverse branch network takes [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]
Figure 6
Figure 6. Figure 6: Schematic illustration of variable-input PI-DIONs architecture. [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]

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Forward citations

Cited by 2 Pith papers

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  2. ELM-DeepONets: Backpropagation-Free Training of Deep Operator Networks via Extreme Learning Machines

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    ELM-DeepONet trains DeepONets by fixing branch and trunk weights and solving a pseudoinverse least-squares problem for a linking matrix, reporting faster and often more accurate results than backprop-trained DeepONets.

Reference graph

Works this paper leans on

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    0.90% 2.51% 8.44% (1,100) 3.52% 1.47% 7.72 % (100,1) 7.55% 10.03% 11.49% Table 5: Approximate training and inference time for PINNs and PI-DIONs. Reaction Diffusion (1e+6 epochs) Darcy Flow (1e+7 epochs) Helmholtz equation (1e+7 epochs) PINNs Training 20m 3h 4h PI-DIONs Training 5h 24h 24h Inference 2ms 5ms 5ms C.2 S ENSITIVITY ANALYSIS The loss function ...

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    14 Published as a conference paper at ICLR 2025 Table 3: Number of trainable parameters for each model. Model Reaction Diffusion Darcy Flow Helmholtz equation DeepONet 6K 70K 72K DeepONet w/ FNO branch 45K 100K 150K FNO N/A 100K 150K PI-DION 12K 100K 110K B.1 R EACTION -DIFFUSION EQUATION : I NVERSE SOURCE PROBLEM USING BOUNDARY MEASUREMENT . Both the rec...

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    Now if K and M are larger than 4R4|Ω|2 log 2/α2 and 4R4|∂Ω|2 log 2/α2, then the following inequality holds. P {xk}K k=1 ∈ Ω, {xj}M j=1 ∈ ∂Ω, {u(i)}N i=1 ∈ U sup η,θ | eLphysics(η, θ) − Lphysics(η, θ,{xk}K k=1, {xj}M j=1)| > 4α ! ≤ 8Ncexp − α2N K 32|Ω|2R4 N + 8Ncexp − α2N M 32|∂Ω|2R4 B . The key part of the proof involves bounding the lefthand side of the ...

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    Both the sensor embedding and position embedding are implemented using simple multilayer perceptrons (MLPs). The final embedding is obtained by computing the inner product of their outputs. For the weights, we used (λ1, λ2) = (1 , 100). In this experiment, we obtained the L2 relative error of 3.83% which is compatible with the original PI-DION results. Th...

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    13 Published as a conference paper at ICLR 2025 A PI-DION S-V0 Figure 5: Schematic illustration of PI-DIONs-v0 architecture. The inverse branch network takes partial measurement data as input and produces the coefficient vector for the target function. The trunk network receives the collocation point x as input and generates the corresponding basis func- ...

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Pith tools

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