{"id":"b76b5a5f-d8e0-46a8-9578-2b709b729752","arxiv_id":"2412.03161","paper_version":2,"verdict":"REJECT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":1,"one_line_summary":"PI-DIONs train DeepONet-style networks to invert PDEs using only physics residuals and partial measurements, with no labeled (measurement, parameter) pairs.","lead":"This paper introduces a neural network method that learns to solve PDE inverse problems, like finding an unknown source or material property from measurements, without needing pairs of correct answers for training. The method embeds the underlying physics equations into the training loss, and the authors claim this removes the need for labeled data while still generalizing to unseen cases.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3's generalization guarantee is unsupported: the stability estimates in §3.1 control error by H^1/H^2 residual norms, while Lphysics is L^2, so small loss does not imply small prediction error.","rationale":"The reader's REJECT verdict is correct. My read agrees with the rejection but identifies a different primary weakness: not the covering-number assumption in Theorems 1–2 (a missing explicit bound, but plausibly repairable by standard compactness/covering-number arguments for bounded-parameter networks), but the loss-to-stability connection on which Theorem 3 depends. The formal stability estimates in Section 3.1 use stronger norms than the loss; the high-frequency example in the concrete test shows Lphysics+Ldata can vanish while the stability RHS does not. The malformed probability expression in Theorem 3 is a downstream symptom of this missing link. The covering-number issue remains a genuine secondary concern: Nc is assumed without any bound or construction, and if Nc is infinite the sample-complexity statements in Theorems 1–2 are vacuous. Both concerns are fatal to the advertised theory, so the verdict stays REJECT; a revision would need to prove a stability estimate for the actual squared L^2 loss or scale the theoretical claims back to what the experiments show.","tokens_in":22243,"tokens_out":12354,"duration_ms":114724,"concrete_test":"On the reaction-diffusion benchmark, take a true pair (u,f) and define u*_n = u + ε_n with ε_n(x,t)=n^{-1/2} sin(nπt/T) φ(x), where φ is smooth and compactly supported in Ω, so the boundary and initial/final data match u. Choose f*_n so that the PDE residual equals ε_n (possible because the operator is linear in f). Then Ldata=0 and Lphysics→0, while the stability RHS in Section 3.1 contains ∥ε_n∥_{H^1(0,T;L2(Ω))} ∼ n^{1/2} → ∞ and ∥f*_n−f∥ does not vanish. This shows small L does not imply small prediction error through the stated stability inequality; to salvage Theorem 3 the authors would need to identify which term in the formal stability estimate is actually controlled by Lphysics.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim—that Theorem 3 guarantees generalization from finite training data—requires a stability estimate connecting the PI-DION loss to the prediction error. That connection is not made. The formal reaction-diffusion stability estimate in Section 3.1 bounds ∥f*−f∥_{L2(Ω)} + ∥u*−u∥_{C([0,T];L2(Ω))} by terms including ∥∂t u* + Δu* − f*g∥_{H^1(0,T;L2(Ω))}, initial/final-data errors with their Laplacians, and an H^2 boundary term. But Lphysics is an L^2 (squared) residual sampled on a grid, and Ldata only constrains the boundary measurement; the H^1 residual norm and the initial/final-data terms are absent. Smallness of an L^2 norm does not imply smallness of an H^1 norm, so the stability RHS is not controlled by the trained loss. The proof of Theorem 3 in Appendix D.3 simply invokes 'stability estimates for a single element' and jumps to error ≤ α; it never bridges this norm gap. The Helmholtz stability bound has the same problem: it uses a Hölder power of the residual and an extra term on Ω\\(Ωm∪Ω0) absent from the loss. The malformed probability expression in Theorem 3 is a visible symptom of this missing link. Thus the advertised finite-sample/finite-grid generalization guarantee is unsupported even in the infinite-sample, infinite-grid limit.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":2154,"tokens_out":4865,"duration_ms":72820,"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":[{"comment":"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.","section":"Section 3.2, Theorem 3 and Appendix D.3"},{"comment":"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.","section":"Section 3.1 versus Section 2 and Appendix D.3"},{"comment":"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.","section":"Appendix D.1, Theorems 1 and 2"},{"comment":"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.","section":"Section 3.3, Proposition 1 and Appendix D.4"},{"comment":"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.","section":"Section 4, Table 1"}],"minor_comments":[{"comment":"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'.","section":"Section 3.1"},{"comment":"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.","section":"Section 3.2, Theorem 1"},{"comment":"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.","section":"Section 3.2"},{"comment":"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.","section":"Appendix C.2, Table 6"},{"comment":"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.","section":"Section 6"}],"recommendation":"reject","confidential_remarks":"The paper is already published as an ICLR 2025 conference paper, but for a journal venue the theoretical issues are not local. The malformed probability expression in Theorem 3 and the L^2-versus-H^1 norm gap between the stability estimates and the loss are fundamental: they concern the core claim that a finite-sample, finite-grid physics-informed loss controls the prediction error. The covering-number and universal-approximation arguments are also not repaired by simple editing. The empirical study shows interesting results, but without unsupervised baselines or error bars it cannot carry the paper on its own. I would not invite a revision unless the authors can supply a corrected theorem that connects the actual loss to the stability estimates, together with a valid proof of the finite-sample generalization statement."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: the PI-DION architecture is a sensible extension of PI-DeepONet to inverse problems, and the experimental section shows it can learn inverse operators without labels. The theory, however, does not hold up. The stability estimates are imported from prior work, and the paper never actually connects them to the training loss.\n\nWhat is genuinely new: the two-branch design (reconstruction plus inverse) sharing a trunk, trained with a physics-informed loss and no labeled target data, is a reasonable and useful idea. The experiments on reaction-diffusion, Helmholtz, and Darcy flow show competitive performance against supervised baselines, including FNO and DeepONet, and the appendix compares fairly to PINNs while highlighting the inference-time advantage. The variable-input extension in Appendix C.3 is a thoughtful addition. The authors also acknowledge the heavy training cost, which is honest.\n\nThe soft spots are serious. Theorem 3's probability expression, as printed, is not a valid probability; it even has a misplaced parenthesis. More fundamentally, the stability estimates in Section 3.1 bound the prediction error by H^1 or H^2 residual norms, while Lphysics is an L^2 residual on a grid. Smallness of an L^2 norm does not imply smallness of an H^1 norm, so the stability estimates do not control the trained loss. The proof in Appendix D.3 just says \"recalling stability estimates for a single element\" and jumps to an error bound; it never bridges that norm gap. The Helmholtz case has the same issue, with an extra term on a subdomain not present in the loss. Theorems 1 and 2 assume finite covering numbers without bounds or a finiteness argument, and the sample-size conditions appear to not depend on epsilon in the right way. Proposition 1 is hand-wavy about derivative approximation, and the proof essentially memorizes the training set rather than showing a general operator approximation. The experiments also lack error bars and are compared mainly against supervised baselines, though the PINN comparison helps.\n\nThe stress-test note is on point: the advertised finite-sample/finite-grid generalization guarantee is unsupported even in the infinite-sample limit, because the loss-to-error connection is missing. This is the central theoretical claim, so the paper as written should not be accepted.\n\nWho is this for? Researchers working on physics-informed operator learning for inverse problems will find the architecture and empirical study useful, even if the theory needs to be ignored or heavily revised. It deserves a serious referee because the empirical idea is viable and the flaws are identifiable and fixable. A revised version that scales back the theoretical claims to a stability heuristic and adds error bars could be a solid contribution. As it stands, I would not rely on the theory.","headline":"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.","tokens_in":23047,"tokens_out":2952,"would_cite":false,"duration_ms":30113,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R30","65M32","68T07"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["inverse problems","operator learning","physics-informed neural networks","stability estimates","partial differential equations","unsupervised learning","DeepONet","generalization bounds"],"falsifier":"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.","tokens_in":22035,"feed_emoji":"📐","tokens_out":8893,"duration_ms":79231,"temperature":0.7,"pith_summary":"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.","feed_headline":"Physics-informed networks learn inverse PDE operators without labels","feed_subtitle":"Stability bounds give finite-sample guarantees for predicting unknown fields from partial measurements.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the stability estimate for the reaction-diffusion inverse source problem that the paper extends to the operator setting.","marker":"Zhang et al. (2023)"},{"why":"Provides the conditional stability estimate for the Helmholtz inverse source problem used in Theorem 3.","marker":"Zhang & Liu (2023a)"},{"why":"Universal approximation theorem for operators that underlies Proposition 1, guaranteeing the loss can be made arbitrarily small.","marker":"Chen & Chen (1995)"},{"why":"Symmetrization technique used in the proofs of Theorems 1 and 2 to bound the gap between empirical and continuous losses.","marker":"Baxter (2000)"},{"why":"DeepONet architecture that PI-DIONs build on and whose universal approximation property is invoked.","marker":"Lu et al. (2021)"},{"why":"Physics-informed DeepONet (PI-DeepONet), the precedent for incorporating PDE residuals into operator training without labels.","marker":"Wang et al. (2021b)"},{"why":"Neural Inverse Operators, the supervised approach PI-DIONs contrast with and aim to beat in experiments.","marker":"Molinaro et al. (2023)"}],"fun_headline_variants":["Physics-informed deep operators invert PDEs without labeled data","Unsupervised PI-DIONs learn inverse PDE operators with stability guarantees","No labels, no problem: Physics-informed operator networks invert PDEs","Label-free inverse PDE solving with physics-informed deep operators","Stability-guaranteed unsupervised learning of inverse PDE operators"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Physics-informed deep operators invert PDEs without labeled data","Unsupervised PI-DIONs learn inverse PDE operators with stability guarantees","No labels, no problem: Physics-informed operator networks invert PDEs","Label-free inverse PDE solving with physics-informed deep operators","Stability-guaranteed unsupervised learning of inverse PDE operators"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000865,"raw_usage":{"total_tokens":3770,"prompt_tokens":984,"completion_tokens":2786,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":600,"completion_tokens_details":{"reasoning_tokens":2702}},"tokens_in":600,"tokens_out":2786,"duration_ms":19502,"temperature":1.0,"reasoning_tokens":2702,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T22:41:48.146294+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Universal approximation theorem for operators that underlies Proposition 1, guaranteeing the loss can be made arbitrarily small."},{"cited_title":"Convolutional neural operators","cited_arxiv_id":null,"evidence_quote":"Neural Inverse Operators, the supervised approach PI-DIONs contrast with and aim to beat in experiments."}],"review_version":1}