{"id":"04de1a6f-aa0b-491f-ab17-f54b080c3043","arxiv_id":"2607.08641","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":6,"one_line_summary":"Constraining neural network partial dependence to match domain-knowledge functional forms during training improves predictive accuracy, data efficiency, and explanation faithfulness on regression problems.","lead":"The paper adds a penalty term to neural network training that forces the model's partial dependence plots to match a known functional shape (e.g., logarithmic or sinusoidal). This improves prediction accuracy and data-efficiency on regression tasks, especially when training data is scarce or out-of-distribution.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"On real-world datasets, performance gains may stem from PD-constraint regularization rather than correct knowledge alignment; Appendix C.2 itself shows mis-specified PD forms still help in extrapolation, undermining the claimed mechanism.","rationale":"The reader correctly identifies the parametric PD assumption as the weakest link but frames it as a forward-looking concern ('if the true PD is not separable...'). The actual evidence for this concern already exists within the paper itself: Appendix C.2 demonstrates that mis-specified knowledge (square root instead of logarithm) yields extrapolation performance comparable to correct knowledge on the Concrete dataset. This is not a hypothetical risk but an empirical finding that partially contradicts the claimed mechanism. The reader's condition ('demonstrate robustness when the PD form is more severely mis-specified') has already been partially met—and the result is unfavorable to the strong interpretation of the central claim. The verdict remains CONDITIONAL because the synthetic experiments are well-controlled and the method is genuinely novel for regression EGL. However, the real-world extrapolation results—the paper's most practically compelling setting—are confounded by the regularization effect, and the paper does not acknowledge or control for this alternative explanation. A clean disambiguation experiment (wrong-shape vs. correct-shape PD on real data, in extrapolation) would settle whether the gains come from knowledge alignment or from regularization. Until then, the claim that interpretations 'align with user-provided knowledge' on real data is not fully established, since the method would produce aligned-looking interpretations regardless of whether the knowledge is correct.","tokens_in":26478,"tokens_out":3478,"duration_ms":248341,"concrete_test":"On the Concrete or PHALK dataset, run linspace constraining with a deliberately wrong-shape PD form (e.g., linear h_phi(age) = phi_0*age + phi_1, or quadratic) and compare extrapolation L_tot against the logarithmic constraint and unconstrained training, across the same training sizes (100/50/25). If the wrong-shape constraint matches or approaches the logarithmic constraint's extrapolation performance, the benefit is regularization, not knowledge alignment, and the central claim's mechanism is unsupported on real data.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim has two parts: (1) constrained models perform better and are more data-efficient, and (2) their interpretations align with user-provided knowledge. On synthetic problems (§4.1, §4.2), both are verifiable because ground truth is known. On real-world datasets (Concrete §4.3, PHALK §4.4), only claim (1) is testable; claim (2) is circular—constraining the PD to be logarithmic and then observing it is logarithmic does not validate the prior. More critically, Appendix C.2 constrains the Concrete dataset with a square-root PD (genuinely mis-specified, since the true relationship is logarithmic) and reports that extrapolation performance is 'very close to that of models constrained with the logarithmic PD,' still outperforms unconstrained training, and that linspace constraining 'reaches performance close to that obtained in the interpolation setting, even though prior knowledge is fundamentally mis-specified.' This directly implies that the extrapolation benefit comes partly from the regularization effect of imposing any smooth constraint on the PD—not specifically from correct domain knowledge. If wrong knowledge helps as much as right knowledge in the regime the paper emphasizes most (extrapolation with scarce data), the mechanism is different from what is claimed, and the real-world performance gains cannot be confidently attributed to correct knowledge alignment rather than to generic regularization. This is the soft spot: the paper's strongest empirical selling point (extrapolation robustness on real data) is also the setting where mis-specification experiments show the knowledge form may not matter.","agreement_with_reader":"partial"},"referee_report":{"model":"glm-5.2","summary":"The paper introduces a method for steering neural network training by constraining the partial dependence (PD) of the model on selected features to align with a known functional form. The authors formulate a penalized objective (Eq. 7) that balances standard mean-squared error with a PD-matching loss, and propose an alternating optimization algorithm (Algorithm 1). The method is evaluated on synthetic regression problems (Friedman, Product), real-world datasets (Concrete, PHALK), and a dynamical systems forecasting task (damped pendulum). The authors compare four configurations (ground-truth vs. approximated prior; in-distribution vs. linspace constraint) against unconstrained training and physics-informed baselines, finding that constrained models are more data-efficient, generalize better under domain shift, and yield PD estimates more faithful to prior knowledge.","tokens_in":26688,"tokens_out":1432,"duration_ms":340656,"significance":"The paper addresses a genuine gap in explanation-guided learning (EGL): most prior work targets classification with feature-importance priors, whereas this work targets regression with functional-form priors on partial dependence. The formulation is clean and the relaxation from hard constraint (Eq. 5) to penalized objective (Eq. 7) is standard and well-motivated. The distinction between in-distribution and linspace constraining is a useful practical contribution, and the extension to dynamical systems (Appendix A.2) broadens applicability. The empirical coverage is commendable: synthetic problems with known ground truth, two real-world datasets, a dynamical system, and a comparison to physics-informed hybrid models (Appendix B). The mis-specification analysis (Appendix C) is a thoughtful addition that few papers in this area include.","major_comments":[{"comment":"Appendix C.2 constrains the Concrete dataset with a square-root PD (genuinely mis-specified, since the assumed true relationship is logarithmic) and reports that extrapolation performance is 'very close to that of models constrained with the logarithmic PD' and that linspace constraining 'reaches performance close to that obtained in the interpolation setting, even though prior knowledge is fundamentally mis-specified.' This directly implies that the extrapolation benefit stems partly from the regularization effect of imposing any smooth constraint on the PD, not specifically from correct domain knowledge. If wrong knowledge helps as much as right knowledge in the regime the paper emphasizes most (extrapolation with scarce data), the claimed mechanism—that performance gains come from correct knowledge alignment—is not fully supported. The authors should either (a) soften the causal claim","section":null},{"comment":"On real-world datasets (§4.3, §4.4), claim (2) that 'interpretations align with user-provided knowledge' is partly circular: the PD is constrained to be logarithmic and then observed to be logarithmic. This is acknowledged implicitly by the absence of L_PD measurements on these datasets. The authors should clarify that on real-world data, only predictive performance is independently verifiable, while PD faithfulness is a consequence of the constraint by construction, not an independent validation of the prior.","section":null},{"comment":"The method assumes the PD of the target function on the selected features is well-approximated by a known parametric form h_phi_k(x_k) (Eq. 3, §3.1). The derivation of Eq. 3 covers purely additive and purely multiplicative cases, but many real-world interactions do not decompose so cleanly. The paper does not discuss what happens when the true PD is not separable into a parametric form of x_k plus a constant. Appendix C shows robustness to mis-specification only for cases where the wrong form still reasonably approximates the true one (Fourier series for sine, square root for logarithm). A brief discussion of the failure mode—when the assumed form is structurally incompatible—would strengthen the paper.","section":null}],"minor_comments":[{"comment":"§3.2, Algorithm 1: the notation for the alternating schedule (s, r) is somewhat opaque. A brief inline comment explaining that S controls the frequency of phi-updates would help readers.","section":null},{"comment":"§4.1, Figure 1: the y-axis label 'L_PD' lacks units/scale context. Figure 4 uses '×10^2' while Figure 1 does not; readers may miscompare across experiments.","section":null},{"comment":"§4.3: the choice of logarithmic PD for age is attributed to Yeh (1998a), but the citation is to a neural network modeling paper. A more direct citation for the logarithmic relationship between concrete age and compressive strength would be appropriate.","section":null},{"comment":"Appendix A.2.1: the redefinition of h_phi_k to map (x_0, x_1) -> (x_1, ...) is introduced abruptly. A sentence explaining that the PD constraint applies to the second component (acceleration) while the first component (velocity) is needed for trajectory integration would improve clarity.","section":null},{"comment":"The term 'linspace constraint' is used throughout but defined only in §3.3. Using it in §4 before readers reach the definition could cause confusion; a forward reference or brief gloss at first use in §4 would help.","section":null},{"comment":"Figure 9 (top row, 25 samples): the PD estimates for unconstrained models are described as 'diverging' but the visual scale makes it hard to distinguish from constrained models. Consider using a shared y-axis range or adding a zoomed inset.","section":null},{"comment":"References: the PHALK dataset citation (Catalan et al., 2025; Batalla et al., 2026) lists a 2026 date for the introductory paper, which appears to be a future date. Please verify.","section":null},{"comment":"§3.3: the distinction between LS and LS_PD is important but the notation is dense. A small worked example showing how LS_PD points are sampled for a 2D PD (e.g., the 25×25 grid mentioned in §4) would aid reproducibility.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The stress-test concern about Appendix C.2 is valid and is the most important issue to address. However, it does not invalidate the paper's contributions: the method is sound, the synthetic experiments do independently validate both performance and PD faithfulness, and the mis-specification results can be framed as a feature (regularization benefit) rather than a bug. The fix is primarily a reframing of the causal claim and a more careful separation of 'correct knowledge helps' from 'any smooth constraint helps.' I would not block publication on this point. The self-citation to Claes et al. (2025) is appropriate as it is a comparison baseline, not a circular validation."},"author_rebuttal":null,"desk_editor":{"model":"glm-5.2","letter":"The paper introduces a new constraint type for explanation-guided learning: instead of constraining feature importance or attention regions (the usual EGL fare, mostly for classification), it constrains the partial dependence of a neural network to match a known parametric functional form. The alternating optimization of network parameters θ and PD-shape parameters φ, plus the linspace-constraint trick for covering poorly sampled regions, is genuinely new. The experiments are thorough—synthetic problems with known ground truth, two real datasets, a dynamical system, and a comparison to physics-informed hybrid models. The method clearly works: constrained models are more data-efficient and produce PD estimates closer to the known functional form, especially with linspace constraints and scarce data. Credit is earned here. The formulation is clean and the relaxation from hard constraint to penalized objective is standard and well-motivated. The reader's verdict of CONDITIONAL is too cautious on the core contribution. The novelty and empirical evidence are solid enough for acceptance. The reader's concern about no shipped code is fair but is a fixable deficiency, not a reason to reject. The stress-test concern about Appendix C.2 is the real soft spot, and it lands. In that experiment, the Concrete dataset's true PD is logarithmic, but constraining it to a square-root form still yields extrapolation performance comparable to the correct logarithmic constraint. The paper itself notes this. If wrong knowledge helps as much as right knowledge in the extrapolation regime—the regime the paper emphasizes most—then the extrapolation benefit partly comes from generic regularization of the PD shape, not specifically from correct domain knowledge. This doesn't invalidate the method, but it muddies the causal story the paper tells. The interpolation results and synthetic experiments (where ground truth is known) are not affected by this concern; the issue is specifically about attributing real-world extrapolation gains to knowledge alignment versus smoothness regularization. A minor point: the alternating optimization has no convergence guarantee, but empirically it seems stable across 10+ seeds. This is a smaller concern than the mis-specification issue. Overall, this is a well-executed paper with a genuinely new EGL constraint type and strong experimental work. The mis-specification finding in Appendix C.2 should be discussed more honestly—acknowledging that part of the extrapolation benefit is regularization would strengthen, not weaken, the paper. Deserves a serious referee.","headline":"PD-constraint method for regression is solid and novel; the extrapolation story has a real soft spot worth addressing","tokens_in":27251,"tokens_out":544,"would_cite":true,"duration_ms":123413,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"glm-5.2","headline":"Constrain the average response, not the full function","keywords":[],"falsifier":"On a regression problem where the true partial dependence on the constrained features is not representable by the chosen parametric family and does not even approximately match it, the constrained model should perform no better than an unconstrained baseline in both prediction error and PD fidelity. The paper's own mis-specification experiment (Appendix C) tests only cases where the wrong form still reasonably approximates the true one.","tokens_in":26723,"feed_emoji":"📐","tokens_out":635,"duration_ms":200098,"temperature":0.7,"pith_summary":"The paper introduces a method called Partial Dependence Steering that constrains a neural network's training so that its partial dependence — the average model response when a subset of input features is varied while all others are held at their observed values — matches a user-specified algebraic form drawn from domain knowledge. The core mechanism is a bi-level optimization: the network parameters and the parameters of the prior PD function are fit in alternating steps, with the network penalized whenever its empirical PD deviates from the prior function. A key design choice is that the constraint points can be drawn either from the training distribution (in-distribution) or from an evenly spaced grid spanning the input domain (linspace). The paper shows that linspace constraints force the network to learn the correct average behavior even in poorly sampled or out-of-distribution regions, producing models that are more data-efficient, generalize better outside the training domain, and yield partial dependence plots that faithfully reflect the injected knowledge. The method is validated on synthetic regression problems, real-world tabular datasets, and a dynamical systems forecasting task, and is compared against physics-informed hybrid models.","feed_headline":"Constrain what a model averages, not what it predicts","feed_subtitle":"A training method that forces neural networks to match known functional relationships in their partial dependence beats unconstrained models","key_machinery":"The partial dependence of a model h on a feature subset x_k is the expectation of h(x_k, x_{-k}) over the marginal distribution of the complementary features x_{-k}. The method minimizes a weighted sum of the standard prediction loss and a PD-matching loss (Eq. 7–8), alternating between updating the network parameters theta and the prior function parameters phi. A separate dataset LS_PD of constraint points x_k can be drawn either from the training distribution or as evenly spaced values across the input domain.","core_discovery":"A neural network can be steered during training so that its partial dependence on selected features matches a known functional form, and when the constraint is applied over an evenly spaced grid rather than only at observed data points, the resulting model is simultaneously more accurate, more data-efficient, and more faithful to the injected knowledge — especially in extrapolation regimes where unconstrained models and in-distribution constraints fail.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Steering neural network partial dependence improves extrapolation and data efficiency","Grid-based partial dependence constraints improve neural network extrapolation","Aligning neural network partial dependence with known functions improves accuracy","Constrained partial dependence yields more accurate and data-efficient models","Forcing models to match known partial dependence improves data efficiency"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The method assumes that the partial dependence of the target function on the selected features is well-approximated by a known parametric form (a function of x_k plus scaling and bias constants). If the true PD involves complex interactions that cannot be captured by such a form, the constraint is mis-specified and may not help.","fun_headline_variants_meta":{"raw":{"variants":["Steering neural network partial dependence improves extrapolation and data efficiency","Grid-based partial dependence constraints improve neural network extrapolation","Aligning neural network partial dependence with known functions improves accuracy","Constrained partial dependence yields more accurate and data-efficient models","Forcing models to match known partial dependence improves data efficiency"]},"model":"glm-5.2","effort":"low","cost_usd":0.0,"raw_usage":{"total_tokens":1316,"prompt_tokens":441,"completion_tokens":875,"prompt_tokens_details":null},"tokens_in":441,"tokens_out":875,"duration_ms":53562,"temperature":1.0,"reasoning_tokens":920,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-10T03:49:58.048684+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"On a regression problem where the true partial dependence on the constrained features is not representable by the chosen parametric family and does not even approximately match it, the constrained model should perform no better than an unconstrained baseline in both prediction error and PD fidelity. The paper's own mis-specification experiment (Appendix C) tests only cases where the wrong form still reasonably approximates the true one.","supporting_citations":[],"review_version":1}