{"id":"b40c038c-5ed7-4084-8a14-3f5af95c757e","arxiv_id":"2506.02363","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A new regression framework estimates the differential operator relating a functional predictor to a functional response, with minimax-optimal rates and a bootstrap goodness-of-fit test.","lead":"This paper proposes a method to learn differential equations that connect two curves from many observed examples, rather than assuming the connection is a simple integral. If the method works, scientists can verify physics laws from data and test whether a proposed formula matches the measurements.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Identification T=L^{-1}∘D∘A requires well-posed L and A; the real-data choice L=d/dlog(p) with two-point Dirichlet BC is ill-posed, so the Section 5 analysis is not a valid instance of the method.","rationale":"Read in good faith, the paper's central claim is that the differential operator D in F=D(U)+ε can be identified via T=L^{-1}∘D∘A and estimated at the minimax rate with a valid bootstrap test. For this claim to hold, T must exist as an operator from V to L2(Ω). The weakest point is whether A and L^{-1} are well-defined for the user-chosen P, B, and L. The paper never states an invertibility or well-posedness condition, and the real-data section chooses L=P=d/dlog(p) with Dirichlet data at both endpoints, for which L^{-1} is only defined on the closed subspace of mean-zero functions. Consequently the transformed model (4) is not equivalent to (1) for typical curves, and the real-data analysis is not a valid instance of the method as stated. The simulation uses well-posed L=-∇² and is internally consistent; the reported rates and bootstrap results may hold under suitable well-posedness assumptions, but those assumptions are unstated. The reader's verdict of CONDITIONAL is supported: the mathematical core can plausibly be repaired by adding explicit conditions on L and by replacing or reworking the real-data example, so I do not see a basis to move to REJECT. The verdict therefore remains unchanged, with the concern sharpened from 'predictors must solve a PDE' to 'the chosen P, B, L must define well-posed boundary value problems.'","tokens_in":20001,"tokens_out":17095,"duration_ms":161422,"concrete_test":"Compute the endpoint differences U_i(6.9)-U_i(6.3) for the 238 ERA5 predictor curves after the preprocessing in Section 5. Under the paper's construction, the zero-Dirichlet solution operator L^{-1} for L=d/dlog(p) requires ∫_{6.3}^{6.9} D_1 U_i = U_i(6.9)-U_i(6.3)=0 for (4) to hold. If these differences are nonzero, attempt to solve L w = D_1(U_i) with w(6.3)=w(6.9)=0 and show no solution exists, demonstrating that the data do not lie in the domain where the transformation is defined.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing gap is the well-posedness condition for the identification D(u)=L T(\\tilde u). Equation (3) defines T=L^{-1}\\circ D\\circ A, which requires A and L^{-1} to be well-defined operators. The paper never states such a condition, and the real-data application violates it. In Section 5, L=P=d/dlog(p) and B=I on both endpoints of (6.3,6.9). For Lu=f with u=0 on Γ, the boundary-value problem u'=f, u(a)=u(b)=0 has a solution only if ∫_a^b f=0; the homogeneous problem has no compact Green's function. Hence L^{-1} is not defined on all of L2(Ω), and T is not an operator from V to L2(Ω). For any observed predictor with U(b)≠U(a), D(U)=U' has nonzero integral, so the identity L∘L^{-1}=Id used after (3) fails and (4) is not equivalent to (1). The simulation uses L=P=-∇² with homogeneous Dirichlet BC, which is well-posed, so the numerical results there do not expose this gap; the real-data p-values in Table 5 are not backed by Theorems 3–4 because the required model (4) is not well-defined for that choice of L.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a new functional regression framework for modeling a function-on-function relation driven by an unknown differential operator D, through the model F = D(U) + ε. To make D estimable, the authors introduce an action-aware identification: assuming each predictor U solves a known PDE P(u)=f, B(u)=g with solution operator A, and choosing an auxiliary linear differential operator L, they define T = L^{-1}∘D∘A and rewrite the model as F = L T(Ũ) + ε. Estimation is performed by penalized least squares over an operator reproducing kernel Hilbert space, with an explicit representer theorem. The paper states a Bahadur representation, proves minimax optimality at rate n^{-r/(r+1)} in the Σ-norm, and establishes asymptotic normality and bootstrap validity for a goodness-of-fit test. Numerical simulations and an ERA5 thermodynamic-energy example illustrate the methodology.","tokens_in":20154,"tokens_out":12482,"duration_ms":112821,"significance":"If the identification step is valid, the paper introduces a genuinely new nonparametric route for learning differential operators from repeated functional data, with statistical guarantees that go beyond symbolic-regression and parametric-ODE approaches. The action-aware reparameterization is elegant, and the minimax lower bound and bootstrap results would be substantial contributions. The paper also provides precise theorem statements and reports reproducible code, which is commendable. However, the applicability of the framework depends critically on the well-posedness of the auxiliary operators A and L^{-1}; the paper does not state these conditions, and the real-data application chooses an L for which they fail. This gap must be addressed before the published claims are fully supported.","major_comments":[{"comment":"The identification step T = L^{-1}∘D∘A is only meaningful if L^{-1} is a well-defined solution operator for Lu=f with u=0 on Γ, but the paper never states an invertibility or well-posedness condition for this boundary value problem. The claim after (3) that 'L∘L^{-1} turns out to be the identity map' is true only on the range of L; when L is not surjective onto L²(Ω), the transformed model (4) is not equivalent to the original model (1), and the estimator and test target a different object. Please add explicit conditions (well-posedness of L and of the PDE in (2), and U belonging to the solution class so that A∘(P,B)=I) and verify them in each application.","section":"Section 2.1, Eqs. (3)-(4)"},{"comment":"The real-data example violates the well-posedness condition just described. With L=P=d/dlog(p) on Ω=(6.3,6.9) and B=I at both endpoints, the BVP u'=f with u(6.3)=u(6.9)=0 has a solution only if ∫_{6.3}^{6.9} f = 0. Observed predictors U are not constrained to satisfy U(6.9)=U(6.3), so D1(U)=U' need not lie in the range of L; then L^{-1}(D1(U)) is undefined and the transformed model (4) is not well-defined. Consequently, the p-values in Table 5 are not covered by Theorems 3-4. This example should be replaced by a well-posed choice of L (e.g., first-order derivative with a single boundary condition), or the framework must be generalized and the theory revised.","section":"Section 5, Table 5"}],"minor_comments":[{"comment":"In the sentence after the definition of D=-∇²-ω², the eigenvalue statement Dφ_k=(kπ)²+ω² is inconsistent with the definition; with φ_k(x)=√2 cos(kπx) one obtains Dφ_k=((kπ)²-ω²)φ_k. Please correct the sign or the definition of D.","section":"Section 4"},{"comment":"The simulation design uses predictor eigenfunctions φ_k that are also eigenfunctions of the true differential operator D, which is a particularly favorable setting for the method. A design where the predictor basis is not aligned with D would give a more demanding test of the action-aware identification.","section":"Section 4"},{"comment":"The kernel K∂ in Example 3 is defined with a Dirac delta δ(x-ξ), which is a distribution rather than a square-integrable kernel; please clarify in what sense this defines a valid operator RKHS, or replace it with an L² kernel.","section":"Example 3, Section 2.2"},{"comment":"The paper states that all technical proofs and auxiliary lemmas are in the Supplementary Material, but this material is not included in the arXiv preprint. Given that Theorem 2's minimax lower bound is a central claim, please make the supplement available with the revision.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The main theoretical idea is promising, but the real-data section is currently an invalid instance of the method because the chosen operator L does not admit the required solution operator L^{-1}. The authors should be asked to fix or replace that example, and the editor should request the supplementary proofs before final acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis paper is a legitimate attempt to push function-on-function regression beyond integral operators to differential relations, and the key idea is genuinely new. Instead of trying to expand the differential operator directly, the authors map each predictor through the solution operator A of a user-specified PDE, define T = L^{-1} D A, and then estimate T in an operator RKHS. That transformation turns the differential problem into a more tractable integral one, and it allows them to state minimax rates and a bootstrap goodness-of-fit test. The theorems are plausible and the simulation study is transparent about its favorable design (predictors are linear combinations of eigenfunctions of D). This is worth reading for anyone working in FDA or operator learning.\n\nThe soft spot is in the real-data example, and it is not minor. In Section 5 they set L = P = d/d log(p) with Dirichlet boundary conditions at both endpoints of (6.3,6.9). For that L, the boundary value problem Lu = f with u=0 on both endpoints has no solution unless ∫ f = 0 over the interval. So L^{-1} is not well-defined on L2(Ω), and the identity L∘L^{-1} = Id that justifies the move from (1) to (4) fails. The p-values in Table 5 are therefore not backed by Theorems 3-4; that analysis is not a valid instance of the method. The simulation uses L = P = -∇² with homogeneous Dirichlet boundary conditions, which is well-posed, so the numerical results do not expose this gap. The paper never states a well-posedness condition on L, a genuine omission.\n\nTwo smaller issues. The proofs are all in a supplementary not included here, so I couldn't verify the chain from the Bahadur representation to the minimax lower bound. And the identification theorem requires that observed predictors exactly satisfy the user-specified PDE; the real-data example assumes that without validation. The real-data result is also essentially a null fit, so it hardly stresses the test.\n\nOverall, the core method is sound if L is well-posed, but the paper overclaims on the real-data front. I would send it to a serious referee, with the explicit request to fix the well-posedness condition and either repair or remove the real-data example. For a reading group, it is a good case study in how a neat theoretical idea can be undercut by an ill-chosen application.","headline":"Novel action-aware identification for function-on-function differential regression, but the real-data analysis is invalid because the chosen L is not invertible.","tokens_in":20794,"tokens_out":7101,"would_cite":false,"duration_ms":54184,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62G05","62G08","62G10","62G20","62F40"],"pacs":[],"model":"deepseek-v4-flash","headline":"An unknown differential operator linking paired functional curves can be estimated at the minimax-optimal rate and checked by a bootstrap test.","keywords":["differential operator","function-on-function regression","functional data","operator learning","reproducing kernel Hilbert space","minimax optimality","goodness-of-fit test","action-aware identification"],"falsifier":"Generate functional data from $F=D(U)+\\varepsilon$ where the predictor curves are solutions of a different differential equation than the assumed $P(u)=f$, $B(u)=g$, then estimate $\\hat D$ by the proposed method and check whether $\\|\\hat D-D\\|_\\Sigma^2$ converges to zero as $n$ grows; failure to converge confirms that the solution-class assumption is necessary, while convergence would show the identification is more robust than stated.","tokens_in":19651,"feed_emoji":"📈","tokens_out":15047,"duration_ms":127397,"temperature":0.7,"pith_summary":"This paper establishes that the unknown differential operator $D$ in a function-on-function regression model $F = D(U)+\\varepsilon$ can be recovered from data, not by expanding $D$ in a basis but by observing its action on functions. The key manoeuvre is to assume each predictor function $U$ solves a user-specified partial differential equation $P(u)=f$ on $\\Omega$ with boundary condition $B(u)=g$ on $\\Gamma$, so that $U$ is represented by its forcing and boundary data through a solution operator $A$; then $D$ is identified with the integral-type operator $T = L^{-1}\\circ D\\circ A$ and estimated by regularized least squares in an operator reproducing kernel Hilbert space (a Hilbert space of operators with continuous evaluation). The estimator attains the minimax rate $n^{-r/(r+1)}$ in the prediction norm $\\|\\cdot\\|_\\Sigma^2$, and a wild-bootstrap goodness-of-fit test is proved asymptotically valid and consistent. An application to the thermodynamic energy equation retains the hypothesized differential operator at the 5% level. The paper thereby turns 'find the differential law from data' into a statistically tractable estimation-and-testing problem.","feed_headline":"Learn differential laws from paired curves at optimal rate","feed_subtitle":"The method turns 'find the differential law' into an estimation and testing problem with proven guarantees.","key_machinery":"The load-bearing object is the action-aware identification $T=L^{-1}\\circ D\\circ A$, where $A$ is the solution operator of the user-specified PDE in (2) and $L$ is a fixed linear differential operator with homogeneous boundary conditions; this rewrites a differential relation as an integral operator so that reproducing kernel Hilbert space tools apply. Estimation is carried out by regularized least squares over an operator reproducing kernel Hilbert space (a Hilbert space of operators with continuous evaluation), and an operator representer theorem reduces the infinite-dimensional optimization to a finite linear system for coefficient matrices in an orthonormal basis. The spectral decay $\\gamma_k\\asymp k^{-r}$ of the covariance operator $\\Sigma$ relative to the operator space sets the regularity parameter $r$ that controls the convergence rate and the test's power boundary.","core_discovery":"The central claim is that a differential regression operator is identifiable through its action. For predictors satisfying the known relation $P(u)=f$, $B(u)=g$, the map $u\\mapsto \\tilde u=(P(u),B(u))$ is invertible on the solution class through the solution operator $A$, so $D$ is in one-to-one correspondence with $T=L^{-1}\\circ D\\circ A$; since $L\\circ L^{-1}$ is the identity, the model becomes $F = L T(\\tilde U)+\\varepsilon$, and inference can target $T$ as an integral operator in an operator reproducing kernel Hilbert space. Minimizing the regularized empirical loss $\\ell_{O;\\lambda}(\\tilde u,f)=\\|f-LO(\\tilde u)\\|^2_{L^2(\\Omega)}+\\lambda\\|O\\|^2_H$ yields an estimator with a Bahadur representation, and choosing $\\lambda\\asymp n^{-r/(r+1)}$ gives the minimax-optimal rate $\\|\\hat T-T\\|^2_\\Sigma=O_P(n^{-r/(r+1)})$. The proposed test statistic $Q_n=n^{-1}\\|S_\\lambda\\tilde\\varepsilon\\|^2$ is asymptotically normal, and the wild bootstrap provides valid critical values, making the test consistent against alternatives in the parametric family $D_\\theta$.","pith_inferences":["If the PDE assumption is violated but the model is still fit, $\\hat D$ may act as a best-fit pseudo-differential operator: it could remain predictive without recovering the true physical relation, so practitioners should verify the solution-class assumption before interpreting recovered operators.","The action-aware identification suggests a natural extension to unknown $P$ and $B$: the PDE itself could be treated as a dictionary selection problem, combining equation discovery with the statistical guarantees developed here.","The same operator-RKHS machinery may extend to nonlinear differential relations by probing the action of $D$ on families of test functions, although the paper treats a linear $L$ throughout.","Applied to other sensor or reanalysis data, the goodness-of-fit test could serve as a falsification tool for candidate physical laws encoded as differential operators, such as Fourier's law or Faraday's law."],"forward_implications":["With $\\lambda\\asymp n^{-r/(r+1)}$, the estimated differential operator achieves the fastest possible prediction-error rate; no estimator based on the same data can beat it asymptotically.","An expert-specified parametric family of differential operators can be checked against data: under the null the bootstrap test keeps its level, and under fixed alternatives the rejection probability tends to 1.","The operator representer theorem makes the method computable: the infinite-dimensional operator estimate reduces to a finite linear system in a chosen orthonormal basis.","The framework applies directly to physical law validation, as illustrated by the thermodynamic energy equation, where the hypothesized derivative operator is retained.","Because $D$ is identified through its action, the method is not limited to a fixed dictionary of differentiation features and can capture derivative information a preset library would miss."],"supporting_citations":[{"why":"Supplies the operator-valued reproducing kernel Hilbert space framework used to define the estimation space.","marker":"Kadri et al. (2016)"},{"why":"Provides the operator RKHS construction and Remark 12 used for the operator-valued kernel and representer theorem.","marker":"Stepaniants (2023)"},{"why":"Source of the spectral decay assumption and the minimax rate benchmark that the estimator matches.","marker":"Cai & Yuan (2012)"},{"why":"Gives the Bahadur representation framework adapted here for operator inference.","marker":"Shang & Cheng (2013)"},{"why":"Inspires the wild bootstrap procedure used to obtain critical values for the goodness-of-fit test.","marker":"Hardle & Mammen (1993)"},{"why":"Provides the Green's function representation of linear solution operators used to construct the solution operator and the transformed operator.","marker":"Lions & Magenes (2012)"},{"why":"Source of the moment and covariance assumptions on noise and covariates adopted in the theory.","marker":"Dette & Tang (2024)"}],"fun_headline_variants":["Learn differential laws at minimax optimal rate","Action-aware differential regression with provable guarantees","From paired curves to differential laws: optimal estimation","Differential operator identification through action and testing","Optimal estimation and testing for differential regression relations"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Predictor functions must be exact solutions of the user-specified PDE $P(u)=f$, $B(u)=g$ with $P$ and $B$ known; if the observed curves are not in this solution class, the estimated operator is not the true differential relation.","fun_headline_variants_meta":{"raw":{"variants":["Learn differential laws at minimax optimal rate","Action-aware differential regression with provable guarantees","From paired curves to differential laws: optimal estimation","Differential operator identification through action and testing","Optimal estimation and testing for differential regression relations"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000174,"raw_usage":{"total_tokens":1312,"prompt_tokens":1004,"completion_tokens":308,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":620,"completion_tokens_details":{"reasoning_tokens":240}},"tokens_in":620,"tokens_out":308,"duration_ms":3831,"temperature":1.0,"reasoning_tokens":240,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T11:26:04.753184+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Generate functional data from $F=D(U)+\\varepsilon$ where the predictor curves are solutions of a different differential equation than the assumed $P(u)=f$, $B(u)=g$, then estimate $\\hat D$ by the proposed method and check whether $\\|\\hat D-D\\|_\\Sigma^2$ converges to zero as $n$ grows; failure to converge confirms that the solution-class assumption is necessary, while convergence would show the identification is more robust than stated.","supporting_citations":[],"review_version":1}