{"id":"9390bd4a-c6ed-4379-a737-63f2f878cf5f","arxiv_id":"2507.18332","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Hi-π combines Buckingham Pi dimensional reduction, multi-branch symbolic regression, and polynomial fitting to rediscover physically meaningful dimensionless parameter combinations and to improve symbolic regression of complex formulas.","lead":"This paper presents Hi-π, a hybrid method that uses dimensional analysis plus symbolic regression to automatically find the dimensionless numbers that control a physical system. The authors test it on Rayleigh-Bénard convection, rough pipe flow, and subsonic compressibility corrections, recovering known groups such as the Rayleigh, Prandtl, and Reynolds numbers.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central claim of basis-independence in Hi-π is asserted but not demonstrated; if extracted groups shift with the arbitrary Buckingham basis, the method does not discover intrinsic parameters.","rationale":"The reader's weakest assumption is exactly the gap I consider most load-bearing. The paper's headline contribution is that Hi-π discovers intrinsic dimensionless parameters, not merely convenient coordinates for fitting. The Buckingham Π step necessarily begins from an arbitrary basis, and the claim that the final extraction is independent of that basis is asserted for one example but never proved or tested. Because the rest of the pipeline is stochastic and finite-sample, the plausible theoretical invariance of the monomial lattice does not automatically guarantee that PySR plus polynomial-order selection returns the same physical groups from different bases. A concrete multi-basis ablation on the pipe-flow case would settle this: if the extracted groups and selected orders are stable across bases, the central claim survives the check; if they vary, the method is not discovering parameters intrinsic to the system but reflecting the user's initial coordinate choice. I do not see a reason to escalate the verdict to REJECT, since the method may well be basis-invariant in practice and the examples are encouraging, but the missing demonstration justifies the CONDITIONAL verdict. Therefore my read does not change the reader's verdict.","tokens_in":14612,"tokens_out":7853,"duration_ms":88781,"concrete_test":"Run the Nikuradse pipe-flow experiment with at least three valid, independent dimensionless bases: (i) the paper's basis in Eq. (18), s1 = Vε/ν and s2 = V²εD/ν²; (ii) the standard basis s1 = Re = VD/ν and s2 = ε/D; and (iii) a non-standard basis such as s1 = VD/ν and s2 = Vε²/(Dν), which is also a full integer basis of the dimensionless lattice. Use identical data, the same train/test split, the same PySR hyperparameters as in Table 2, the same 1st-to-8th order polynomial scan, and at least 10 random seeds per basis. Then compare, for each basis, the extracted parameter combinations rewritten in original variables, the order-loss curves, the selected polynomial order, and test R².","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim is that Hi-π extracts intrinsic dimensionless groups from data, including multiple groups simultaneously. This requires that the arbitrary choice of dimensionless basis vectors w_b in Eqs. (9) and (18) does not determine the output. The paper only asserts for pipe flow that 'you can also select other dimensional index matrices w_b, and the final result is the same' (Section 3.3, Figure 7 caption) and gives no proof, theorem, or systematic ablation. In log space, a change of dimensionless basis is a unimodular integer transformation s' = A s, so for infinite data and an exact symbolic search the space of monomial combinations is unchanged, which makes the claim plausible. But the actual pipeline uses stochastic PySR with a fixed symbol library, finite samples, and a manually selected polynomial order, and the order-loss tradeoff can prefer different coordinate expressions for different bases. No argument rules out this basis dependence. Since the extracted combinations are the delivered scientific output, the assertion that the method 'can find the low-dimensional parameters inherent in the system itself' (Section 3.2) rests on an untested invariance. If the output changes with the initial basis, Hi-π is a basis-dependent reparametrization of the data rather than a discoverer of intrinsic parameters. This is the load-bearing weak point of the central claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes Hi-π, a physics-data hybrid method that combines Buckingham-Pi dimensional analysis, multi-branch symbolic regression (PySR), and polynomial regression to extract dimensionless parameter combinations from data. The method is demonstrated on three synthetic mathematical examples and on three fluid-mechanics problems: Rayleigh-Bénard convection (where it claims to rediscover the Rayleigh and Prandtl numbers), rough circular pipe flow (where it claims to rediscover the Reynolds number and relative roughness), and subsonic compressibility correction (where it recovers the Prandtl-Glauert factor and uses it to improve symbolic-regression discovery of the Karman-Tsien formula). The central claim is that the pipeline can find the low-dimensional intrinsic parameters of a physical system and can extract multiple parameter combinations simultaneously, overcoming limitations of power-law-only methods and single-branch symbolic-regression approaches.","tokens_in":14847,"tokens_out":5602,"duration_ms":58745,"significance":"If the central claims hold, Hi-π would be a useful, interpretable tool for data-driven dimensional analysis, particularly because it handles multiple dimensionless groups and nonlinear parameter combinations without a power-law ansatz. The mathematical examples are well designed and the use of external Rayleigh-Bénard and Nikuradse pipe-flow data gives the application section a realistic character. However, the paper's load-bearing assertions rest on two points that are currently not established: the claimed independence of the results from the arbitrary dimensionless-basis choice, and the quantitative reliability of the physical validations. The manuscript also does not provide code, data, or a precise definition of the success metrics used in the symbolic-regression comparisons, which limits reproducibility. The idea is coherent and the mathematical benchmarks appear to verify the implementation, but the physical claims need stronger evidence before the method can be accepted as a discoverer of intrinsic dimensionless parameters.","major_comments":[{"comment":"The central claim that Hi-π discovers 'intrinsic' dimensionless parameters requires that the output be independent of the arbitrarily selected dimensionless basis vectors w_b, but this is only asserted ('you can also select other dimensional index matrices w_b, and the final result is the same') and is neither proved nor systematically demonstrated. In log space, a change of basis is a unimodular integer transformation s' = A s, and while the monomial span is invariant in the exact infinite-data limit, the actual pipeline uses stochastic PySR, finite samples, and a manually selected polynomial order, so the order-loss trade-off can in principle favor different coordinate expressions for different bases. Please provide either a proof under the stated assumptions or an ablation study varying w_b over multiple random seeds for the mathematical, pipe-flow, and Rayleigh-Bénard cases, reporting the extracted combinations and test errors.","section":"§3.3, Eq. (18), Fig. 7"},{"comment":"The Rayleigh-Bénard validation is not quantitatively assessable. The text states that 'incompatible transition state data' were deleted and that the Prandtl range was used to split datasets, but it does not report the resulting sample sizes, the exclusion criteria, the exact symbolic expressions returned by Hi-π, or how those expressions were identified as the Rayleigh and Prandtl numbers. Figure 6 shows prediction comparisons without numerical errors, error bars, or repeated-run variability. The authors should report the discovered expressions in a table, describe the identification procedure, and give quantitative metrics such as RMSE and R² for interpolation and extrapolation for both Hi-π and PyDimension.","section":"§3.2, Fig. 6"},{"comment":"The claim that Hi-π outperforms SFL on multi-parameter examples lacks a defined success criterion. The paper refers to 'extraction accuracy' and shows qualitative results, but it does not define how an extracted expression is judged correct (exact string match, symbolic equivalence after simplification, or numerical tolerance), how many repeated trials were used, or the per-example success rates. Without these details, the comparison and the claimed robustness to noise cannot be independently evaluated. Please specify the metric, the number of random seeds, and the per-example success rates for both methods.","section":"§3.1, Fig. 4"},{"comment":"The knowledge-discovery comparison reports accuracy rates (e.g., 70% for Hi-π and failure for PySR) without defining how the symbolic expressions are compared with the target formula, the noise level, or the number of trials. This is a load-bearing claim because it supports the paper's conclusion that Hi-π helps discover hierarchical expressions. Please define the accuracy metric, the data-generation noise, the trial count, and provide representative discovered expressions for both the Prandtl-Glauert and Karman-Tsien cases.","section":"§3.4, Fig. 9"}],"minor_comments":[{"comment":"The name 'Rayleigh-Bernard' should be written as 'Rayleigh-Bénard' in the abstract, the introduction, and Section 3.2.","section":"Throughout"},{"comment":"The sentence 'Saha et al. proposed HiDeNN' is cited as reference [19], but reference [19] is the paper by Evangelou et al.; the HiDeNN paper appears to be reference [21]. Please correct this citation mismatch.","section":"§1, references [19] and [21]"},{"comment":"Equation (10) uses the symbol k for thermal diffusivity while the surrounding text uses κ; please unify the notation.","section":"Eq. (10)"},{"comment":"The text refers to 'DimenisonNet'; this appears to be a typo for 'DimensionNet', and the cited reference [21] is listed as HiDeNN, so the terminology and reference need to be aligned.","section":"§3.3"},{"comment":"The iterative strategy for determining the number of dimensionless parameter combinations is described only in prose at the end of Section 3.2; a pseudocode or algorithm box would substantially improve reproducibility.","section":"§3.2, iterative strategy"},{"comment":"The data availability statement says data will be made available on request; given the stochastic nature of the symbolic-regression search, depositing the code and datasets would materially strengthen the paper.","section":"Data availability"}],"recommendation":"major_revision","confidential_remarks":"The manuscript's main novelty relative to the authors' earlier SFL work [23] is the multi-branch extraction of several parameter combinations and the use of the extracted combinations as a preprocessing step for symbolic-regression knowledge discovery. The comparison with SFL is therefore important, and the relationship should be stated explicitly rather than implicitly. I would also encourage the editor to require the authors to make code and data available, since the quantitative claims currently rest on unreported experimental details."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The genuinely new piece in Hi-π is the multi-branch symbolic regression step: it extracts several dimensionless parameter combinations at once, rather than the single combination that prior methods like SFL or PyDimension return. The second clever piece is to feed those extracted combinations back into symbolic regression as transformed inputs, which helps recover hierarchical formulas like the Prandtl-Glauert correction. The mathematical examples verify the multi-branch extraction works and that it is robust to 5% noise; the Rayleigh-Bénard and pipe-flow examples recover the expected textbook groups; the compressibility example shows the hierarchical discovery idea can beat direct PySR on a deliberately hard synthetic formula. The paper is clearly written and the literature survey is useful.\n\nThe soft spots are in proportion. The biggest one is the basis-independence claim in Section 3.3 and Figure 7: the authors say choosing other dimensional index matrices w_b gives the same final result, but they provide no proof, theorem, or ablation. In exact symbolic search the function space is invariant under unimodular reparametrization, but the actual pipeline uses stochastic PySR, finite samples, a fixed symbol library, and a complexity penalty that can prefer different coordinate expressions for different initial bases. Without a systematic test across multiple random bases, the central claim that the method discovers intrinsic parameters of the system is not fully supported. This is fixable but it is load-bearing.\n\nThe physical validations also lean qualitative: there are no error bars, no precise quantitative comparison against PyDimension or SFL, and the Rayleigh-Bénard section excludes transition-state data without showing the method is stable under that choice. The pipe-flow section uses a limited Nikuradse dataset and the order-selection logic is reasonable but somewhat heuristic. Finally, the paper says data will be made available on request, but for a methods paper built on stochastic symbolic regression, releasing code and data with exact PySR settings is close to necessary for reproducibility.\n\nNone of this is a fatal flaw. The method is plausible, the examples are encouraging, and the hierarchical 'extract-then-discover' step has real practical value. What the paper needs is a demonstration of basis-independence, quantitative comparisons with uncertainty, and released artifacts. I would send it to a serious referee; it is not a desk reject.","headline":"A sensible incremental method for extracting multiple dimensionless groups via multi-branch symbolic regression, but the paper's central claim of basis-independence is asserted without proof and needs demonstration before the 'intrinsic parameters' narrative holds.","tokens_in":15360,"tokens_out":4135,"would_cite":false,"duration_ms":48327,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A three-stage pipeline that combines Buckingham-Pi dimensional analysis, multi-branch symbolic regression, and polynomial regression can rediscover a system's intrinsic dimensionless parameter combinations from data, the paper argues.","keywords":["symbolic regression","parameter reduction","dimensional analysis","data-driven discovery","dimensionless numbers","Buckingham Pi theorem","Rayleigh–Bénard convection","pipe flow"],"falsifier":"Run the circular-pipe example with a different valid basis, for instance $w_{b1}' = (1,-1,1,0)^T$ and $w_{b2}' = (0,0,-1,1)^T$, which yield $\\mathrm{Re} = VD/\\nu$ and $\\varepsilon/D$, and apply the Hi-π pipeline to the same experimental pipe-flow data. If the extracted optimal combination at the selected polynomial order is no longer $\\mathrm{Re}$ and $\\varepsilon/D$, or if the order-loss trade-off curve changes shape, then the claimed intrinsic extraction fails.","tokens_in":1656,"feed_emoji":"🌊","tokens_out":2190,"duration_ms":85297,"temperature":0.7,"pith_summary":"Hi-π is a three-stage pipeline: dimensional analysis first compresses the raw physical inputs into an arbitrary set of dimensionless groups, symbolic regression then proposes several dimensionless parameter combinations using a multi-branch tree, and polynomial regression scores each combination by predictive accuracy. The paper claims that this pipeline can rediscover the physically meaningful dimensionless numbers of a system without the user knowing them in advance, including multiple groups at once. Across the tested cases, it recovers the Rayleigh and Prandtl numbers for Rayleigh–Bénard convection, the Reynolds number and relative roughness for circular-pipe flow, and a form of the compressibility correction for subsonic flow. A sympathetic reader would care because this offers a data-driven route to the key insight of dimensional analysis: finding the few combinations of variables that actually organize a physical problem.","feed_headline":"Hi-π rediscovers the dimensionless numbers of fluid physics","feed_subtitle":"A three-layer pipeline extracts Rayleigh, Prandtl, Reynolds and roughness groups, plus the compressibility correction, from data.","key_machinery":"The carrying object is a composition of three maps: a Buckingham-Pi dimensional embedding that sends the physical inputs $\\boldsymbol{p}$ to an arbitrary set of dimensionless variables $\\boldsymbol{s}$; a multi-branch symbolic-regression tree that searches for $n$ parameter combinations $\\pi_i = g_i(\\boldsymbol{s})$ without restricting them to power-law products; and a multivariate polynomial regression $F(\\boldsymbol{\\pi}, \\boldsymbol{\\beta})$ whose squared error is the loss that drives the symbolic search. The multi-branch tree structure is what allows several parameter combinations to be extracted simultaneously, and the polynomial order serves as an explicit complexity knob: the paper selects the 'best' parameter combination by looking for the order at which the loss stops improving, balancing accuracy against interpretability.","core_discovery":"The central claim is that a hierarchy of dimensional reduction, unconstrained symbolic search, and polynomial mapping can extract the intrinsic dimensionless parameter combinations of a physical system from data, even when several such combinations matter simultaneously. In the cases presented, the extracted combinations coincide with the standard ones: the Rayleigh number and Prandtl number for Rayleigh–Bénard convection, the Reynolds number and relative roughness for rough circular-pipe flow, and the incompressible pressure coefficient together with the Prandtl–Glauert factor for the Karman–Tsien compressibility correction. The paper further claims that this extraction is robust to the data range and to differences in sensitivity between parameter combinations, whereas single-combination extraction methods are not.","pith_inferences":["If the basis-independence claim holds generally, Hi-π can be understood not just as a rediscovery tool but as a data-driven way to select a canonical coordinate system on the space of dimensionless groups, which would connect it to parameter-identifiability and active-subspace ideas.","A natural test beyond the paper's four examples would be a system whose true dimensionless group is not already known from classical theory; success there, rather than in rediscovering known groups, would be the stronger evidence that the method finds intrinsic structure.","Because the paper notes that polynomial mappings may be limited in high-dimensional complex problems, replacing the polynomial scoring layer with a model-free measure such as mutual information is a plausible extension that could broaden the method's range.","The claim that extracted groups are 'intrinsic' depends on the output quantity chosen: rerunning the pipeline with a different target quantity would likely produce different optimal parameter combinations, so the method discovers groups that characterize a specific input-output relationship rather than the system in isolation."],"forward_implications":["A user can start from any valid set of dimensionless groups produced by Buckingham-Pi analysis and still recover the conventional physical groups, if the claimed independence from the initial basis holds.","Multi-branch symbolic regression can find physically essential but weak dimensionless numbers alongside a dominant one, which is a case where single-combination methods fail.","Feeding the discovered parameter combinations into symbolic regression as transformed variables materially increases the success rate of discovering complex hierarchical formulas such as the Karman–Tsien correction.","The iterative strategy of increasing the number of branches until prediction error stops improving gives a data-driven estimate of the intrinsic dimension of a system's low-dimensional manifold.","Comparing the polynomial-order-versus-loss curve across candidate parameter combinations provides a complexity-aware criterion for selecting the most physically interpretable dimensionless representation."],"supporting_citations":[{"why":"Supplies the symbolic-regression engine with the multi-branch tree structure used for simultaneous extraction of parameter combinations.","marker":"[26]"},{"why":"The single-combination data-driven dimensionless-learning baseline whose results depend on data range, against which Hi-π is compared in the Rayleigh–Bénard example.","marker":"[6]"},{"why":"The referenced method for balancing accuracy and complexity in the pipe-flow example, using a BIC-style threshold rather than polynomial order.","marker":"[21]"},{"why":"Provides the experimental rough-pipe data set used to rediscover the Reynolds number and relative roughness.","marker":"[33]"},{"why":"Three experimental Rayleigh–Bénard data sets used to validate multi-scale unified representation and extraction of Ra and Pr.","marker":"[29-31]"},{"why":"Supplies the Sobol variance decomposition used to rank the importance of the extracted parameter combinations.","marker":"[27]"},{"why":"Demonstrates that variable transformation before symbolic regression aids equation discovery, with Hi-π automating that transformation.","marker":"[38]"}],"fun_headline_variants":["Hi-π auto-extracts key dimensionless numbers from fluid data","Hierarchical learning recovers Rayleigh, Prandtl, and Reynolds numbers","Physics-data hybrid discovers classic fluid dimensionless groups","New method finds dimensionless parameters in fluid flows"],"cache_read_input_tokens":17536,"weakest_assumption_plain":"The final extracted dimensionless parameter combinations are independent of which valid set of dimensionless basis vectors is chosen at the Buckingham-Pi step; the paper asserts this for the pipe-flow example but supplies no proof or systematic demonstration.","fun_headline_variants_meta":{"raw":{"variants":["Hi-π auto-extracts key dimensionless numbers from fluid data","Hierarchical learning recovers Rayleigh, Prandtl, and Reynolds numbers","Physics-data hybrid discovers classic fluid dimensionless groups","New method finds dimensionless parameters in fluid flows"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000206,"raw_usage":{"total_tokens":1359,"prompt_tokens":868,"completion_tokens":491,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":484,"completion_tokens_details":{"reasoning_tokens":426}},"tokens_in":484,"tokens_out":491,"duration_ms":5173,"temperature":1.0,"reasoning_tokens":426,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T18:14:47.659658+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the circular-pipe example with a different valid basis, for instance $w_{b1}' = (1,-1,1,0)^T$ and $w_{b2}' = (0,0,-1,1)^T$, which yield $\\mathrm{Re} = VD/\\nu$ and $\\varepsilon/D$, and apply the Hi-π pipeline to the same experimental pipe-flow data. If the extracted optimal combination at the selected polynomial order is no longer $\\mathrm{Re}$ and $\\varepsilon/D$, or if the order-loss trade-off curve changes shape, then the claimed intrinsic extraction fails.","supporting_citations":[{"cited_title":"Data-driven discovery of dimensionless numbers and governing laws from scarce measurements[J]","cited_arxiv_id":null,"evidence_quote":"The single-combination data-driven dimensionless-learning baseline whose results depend on data range, against which Hi-π is compared in the Rayleigh–Bénard example."},{"cited_title":"Hierarchical deep learning neural network (HiDeNN): an artificial intelligence (AI) framework for computational science and engineering[J]","cited_arxiv_id":null,"evidence_quote":"The referenced method for balancing accuracy and complexity in the pipe-flow example, using a BIC-style threshold rather than polynomial order."},{"cited_title":"Laws of flow in rough pipes[J]","cited_arxiv_id":null,"evidence_quote":"Provides the experimental rough-pipe data set used to rediscover the Reynolds number and relative roughness."},{"cited_title":"Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates[J]","cited_arxiv_id":null,"evidence_quote":"Supplies the Sobol variance decomposition used to rank the importance of the extracted parameter combinations."},{"cited_title":"AI Feynman: A physics -inspired method for symbolic regression[J]","cited_arxiv_id":null,"evidence_quote":"Demonstrates that variable transformation before symbolic regression aids equation discovery, with Hi-π automating that transformation."}],"review_version":2}