{"id":"7ce9fb45-6cec-41e9-af4e-2ab9e2d00a45","arxiv_id":"2608.09483","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"HRE-PINN represents PDE solutions with a hierarchical tensor network and sparse rank-evolving vectors, reporting accuracy gains over SPINN, TT-PINN, and Tucker-PINN across five benchmarks.","lead":"This paper introduces HRE-PINN, a physics-informed neural network whose solution representation is a hierarchical tensor network with ranks that shrink during training. The authors report lower errors than three tensor-based PINN baselines on five PDE benchmarks.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Automatic rank determination is not well-posed: scale degeneracy lets the L1 penalty shrink without changing the represented function, and no exact-zero/thresholding mechanism is specified (Sec. 3.2–3.3).","rationale":"The reader's weakest assumption correctly identifies that L1 regularization plus Adam may not yield exact zeros. My stress-test goes deeper: the L1 objective is not merely hard to optimize, it is not even well-posed for rank determination because of a scaling degeneracy. Since r vectors are the only penalized parameters and the univariate networks and core tensors are free, any nonzero r can be scaled down by an arbitrary positive factor while the represented function is preserved by inverse scaling of the factors. This means the L1 penalty does not select a meaningful r scale, and the support of r is not robustly defined in floating-point arithmetic. This directly threatens the central novel claim of automatic rank determination, not just the empirical evaluation. The architecture may still be valuable as a flexible tensor-network PINN, and the claimed performance could survive once normalization or a proximal thresholding scheme is added; the ablation studies are internally coherent but do not probe the scale ambiguity or exact-zero behavior. For these reasons the appropriate verdict remains CONDITIONAL, matching the reader, with the scale-degeneracy issue added as a required fix.","tokens_in":18866,"tokens_out":6561,"duration_ms":79426,"concrete_test":"Run one controlled experiment on the 5D Poisson benchmark (Table 2): log the final values of all r_in and r_out vectors and count entries that are exactly zero at machine precision. Then re-run the same experiment with all r entries initialized at 1.0 versus 1e-3, renormalizing the univariate networks and core tensors so the initial HRE function is identical. If no entries are exactly zero, or if the recovered rank (number of nonzero entries, using any small threshold) differs across the two initializations, the automatic rank-determination claim fails. An analytical companion check is to verify whether Eq. (3) is invariant under r -> αr with compensating core/network scaling.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central novelty is automatic rank determination. In the HRE representation (Definition 2), each outer entry r_out_n(i) multiplies g_n(x_n)(i). If every entry of r_out_n is scaled by α and the output of g_n is divided by α (absorbed into the final-layer weights of the univariate network), the represented function is unchanged but ||r_out_n||_1 scales by α. Similarly, each inner r_in_{k,l}(j) can be counter-scaled in an adjacent FCTN core. Thus, for any representable target, the L1 penalty in Eq. (3) can be made arbitrarily small over equivalent representations, so the magnitude of r carries no information about rank. The paper never normalizes g_n or the cores, never applies a proximal/hard-threshold step, and specifies no threshold. Algorithm 1 simply runs Adam on the nonsmooth objective. Section 3.3's claim that 'the number of non-zero entries ... directly gives the suitable rank' therefore does not follow: Adam subgradient updates rarely produce exact zeros in floating point, and any threshold would reintroduce the manual hyperparameter choice the paper claims to eliminate. Because automatic rank determination is contribution 2 and the basis for 'freeing us from manual rank tuning', the central claim is not yet established. The issue is fixable, but the paper as written lacks the required constraint or thresholding mechanism.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a hierarchical rank-evolving (HRE) representation for multivariate functions and embeds it in a physics-informed neural network (HRE-PINN). HRE combines an outer function Tucker decomposition, whose univariate basis functions are neural networks, with an inner fully-connected tensor network (FCTN) decomposition of the core tensor. Each edge of the inner network and each outer component is modulated by a learnable vector, regularized with an L1 penalty so that redundant modes are purportedly pruned during training, yielding automatic rank and structure discovery. The total loss combines PDE residual, boundary, and sparsity terms. Experiments on five PDEs (3D Helmholtz, 5D Poisson, (2+1)D Klein-Gordon, flow mixing, Navier-Stokes vorticity) compare HRE-PINN against SPINN, TT-PINN, and Tucker-PINN, reporting consistently lower RMSE and relative L2/L-infinity errors. Ablation studies on the 5D Poisson equation attribute gains to the hierarchical design, the rank-evolving mechanism, and the customized inner structure.","tokens_in":19137,"tokens_out":6983,"duration_ms":69229,"significance":"If the claimed results are reproducible, the paper would offer a meaningful step toward adaptive tensor representations in PINNs: the HRE structure is a natural extension of FCTN and SVDinsTN into continuous function space, and the experimental suite covers useful hard cases. The work provides explicit definitions and an algorithm, plus ablation studies that isolate components. However, the central novelty—automatic rank determination via L1-regularized rank-evolving vectors—is currently under-justified, and the empirical evidence is weakened by test-set model selection, missing baselines, and absent error bars. With fixes, the method could be a solid contribution to scientific machine learning.","major_comments":[{"comment":"The sentence in Section 4 that 'The model checkpoint yielding the best prediction accuracy on test samples is retained to produce the final results' implies that the reported errors in Tables 1-5 are obtained by test-set model selection. This biases all reported accuracies upward and makes the claimed 'consistently outperform' unverifiable as a generalization statement. Use a validation set (disjoint from the test set used for reporting) or an early-stopping criterion based on training loss, and report the test errors computed at the selected checkpoint. In addition, because PINN training is stochastic, please report means and standard deviations over at least 3-5 random seeds.","section":"Section 4, test-set checkpoint selection"},{"comment":"The comparison omits the most relevant baselines for the claimed contribution: a functional FCTN-based PINN and a functional SVDinsTN-based PINN (the structure-search method of Ref. [48] that HRE builds on). A standard (non-tensor) PINN is also not included. Since the inner layer of HRE is an FCTN decomposition with rank-evolving vectors, and since SVDinsTN is precisely a structure-search tensor network, these baselines are needed to support the abstract's claim that HRE-PINN 'consistently outperform[s] existing state-of-the-art approaches.' Without them, the improvements in Tables 1-5 could be due to the extra flexibility of the FCTN backbone rather than the rank-evolving mechanism.","section":"Section 4, baseline selection"},{"comment":"The loss in Eq. (3) involves hyperparameters eta, lambda, and mu, and the method requires initial outer widths I_n, inner bond dimensions J_{k,l}, univariate network architectures, and optimizer settings. None of these are reported per experiment; the text merely says baselines are 'configured following the original publications' and 'consistent network capacities.' This is insufficient for reproducibility and makes the claim of freeing users from 'manual rank tuning' hard to assess, since many other manual choices remain. Please provide a table of all hyperparameters and training budgets for every benchmark.","section":"Section 4, Eq. (3), hyperparameters"},{"comment":"In Section 5.3, Table 8 compares 'Tensor directly', TT, TR, and FCTN with the 'Proposed' structure, but the fixed-structure baselines are presumably run without the inner rank-evolving vectors, whereas the Proposed row includes them. The attribution of the accuracy gain to the underlying structure is therefore confounded with the gain from rank evolution. Run the fixed structures with the same outer rank-evolving projection and report whether inner rank-evolving vectors are enabled in each row, or redesign the ablation so that only the inner graph topology varies.","section":"Section 5.3, Table 8"}],"minor_comments":[{"comment":"The index range '1 <= k <= l <= N' for the inner rank-evolving vectors is ambiguous and likely should be '1 <= k < l <= N', since FCTN edges connect distinct factors.","section":"Definition 2, Section 3.2"},{"comment":"The boundary loss term is weighted by eta times 1/N_B, while the residual loss has no explicit outer weight; this asymmetry may bias the optimization and should be clarified.","section":"Eq. (3), Section 3.3"},{"comment":"In the PDE system, the boundary condition line reads 't in Omega' in the second line; this appears to be a typo and should be 't in Gamma' to match the time interval.","section":"Section 4.3, Klein-Gordon equation"},{"comment":"The symbol b is used both for the spatially varying coefficient b(x,y) in the advection terms and for the boundary function b(x,y,t); please rename one of them to avoid confusion.","section":"Section 4.4, flow mixing equation"},{"comment":"There are typos such as 'underling structure' (end of Section 3.2), 'reveal the underlying structure' for 'reveals' (Section 3.3), and 'produces' for 'produce' (Section 4.5); a careful proofread is recommended.","section":"Throughout"},{"comment":"The caption of Figure 1 does not explain the meaning of nodes and edges in the tensor network diagrams; a legend would help readers understand the structural differences.","section":"Figure 1"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is authored by members of the group that developed FCTN and SVDinsTN, and the proposed HRE representation is essentially a functional adaptation of those decompositions. The absence of FCTN-PINN and SVDinsTN-PINN baselines in the experiments is therefore particularly noticeable. I would recommend requiring the authors to include those baselines and to clarify the novelty relative to their prior work. The paper also lacks a limitations discussion; a section acknowledging the thresholding issue and the remaining hyperparameters would be appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper proposes HRE-PINN, a tensor-based representation for PDE solutions that nests an FCTN-style inner tensor network inside a function Tucker outer layer, with learnable sparse rank-evolving vectors on both levels. That hierarchical combination is new, and it targets two real pain points in the T-PINN literature: manual rank tuning and fixed tensor topologies. The ablation studies on the 5D Poisson equation are well-constructed and show each component contributes, and the reported gains over SPINN, TT-PINN, and Tucker-PINN are consistently large, including on the tough Navier-Stokes case. The architecture is coherent and the writing is clear.\n\nBut the central claim — that ranks and structure are automatically determined by L1 regularization — does not survive scrutiny. The representation has a scale degeneracy: scaling r_out_n by α and dividing the output of g_n by α leaves the represented function unchanged but shrinks ||r_out_n||_1. The same holds for the inner rank-evolving vectors via adjacent FCTN cores. So the L1 penalty can be driven arbitrarily close to zero without pruning anything meaningful. The paper never normalizes the factors, never applies a hard-threshold or proximal step, and Section 3.3's assertion that non-zero entries after training directly give the suitable rank is unjustified. Any threshold you add reintroduces the manual hyperparameter the paper claims to eliminate.\n\nThe empirical section has additional, addressable problems: the final checkpoint is selected using test samples, there are no error bars, and natural baselines like functional SVDinsTN and FCTN-PINN are missing. No code or data is released. These are fixable in revision, but as reported the performance numbers are not fully trustworthy.\n\nDespite the flaw, this is not a paper to wave away. The representation idea is clever, the experiments span diverse PDEs, and the results suggest something real even if the mechanism is not what the authors describe. A serious referee should engage with it and insist on either a constraint that fixes the scale degeneracy (e.g., normalizing the rank vectors) or a transparent thresholding protocol with sensitivity analysis. If that gets fixed, the paper could be a solid contribution to the T-PINN literature.\n\nMy recommendation: send it to peer review, but flag the rank-determination mechanism as the load-bearing issue.","headline":"A genuinely new hierarchical tensor representation for PINNs, but the automatic rank determination claim collapses under a scale-degeneracy argument; worth refereeing if the authors fix the mechanism.","tokens_in":19708,"tokens_out":3006,"would_cite":false,"duration_ms":32472,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["15A69","68T07"],"pacs":[],"model":"deepseek-v4-flash","headline":"HRE representation self-determines tensor ranks and structure during training, and the resulting HRE-PINN outperforms fixed-rank tensor-based PINNs across five PDE benchmarks.","keywords":["physics-informed neural networks","tensor network decomposition","rank-evolving representation","automatic rank determination","multivariate function approximation","sparsity regularization","high-dimensional PDEs","fully-connected tensor network"],"falsifier":"Train HRE-PINN on the 5D Poisson problem and inspect the trained values of $r^{(\\mathrm{in})}$ and $r^{(\\mathrm{out})}$: count how many entries are exactly zero in floating point, and then vary the sparsity weights $\\lambda$ and $\\mu$ slightly and see whether the reported rank (the number of nonzero entries) changes discontinuously or drifts. If no exact zeros appear without a thresholding step, or if the revealed rank is highly sensitive to the penalty weight, the automatic rank determination claimed in the paper fails.","tokens_in":18605,"feed_emoji":"🧮","tokens_out":9169,"duration_ms":84689,"temperature":0.7,"pith_summary":"The paper claims that the right tensor representation for a multivariate solution can be found by the training process itself, not fixed in advance. Its hierarchical rank-evolving (HRE) representation writes a function as an outer sum of univariate networks multiplied into a small core tensor, and then expands that core tensor as a fully-connected tensor network whose internal edge sizes are controlled by learnable sparse vectors. An $\\ell^1$ penalty on those vectors is meant to drive redundant entries to zero, so the ranks and the network topology are revealed automatically, eliminating manual rank tuning. The resulting HRE-PINN is reported to beat SPINN, TT-PINN, and Tucker-PINN on every one of the five benchmark PDEs tested (3D Helmholtz, 5D Poisson, (2+1)D Klein-Gordon, flow mixing, and Navier-Stokes), often cutting the best baseline error by roughly 40 percent to 80 percent.","feed_headline":"Rank-evolving tensor nets auto-select ranks and beat fixed-rank PINNs","feed_subtitle":"Sparse learnable weights reveal the tensor network's true size during training, cutting errors by up to 80 percent.","key_machinery":"The load-bearing object is the HRE representation: an outer function-Tucker expansion in which each univariate network $g_n(x_n)$ is weighted by a learnable vector $r_n^{(\\mathrm{out})}$ before being contracted with a core tensor $\\mathcal{T}$, and an inner decomposition of that core tensor as a fully-connected tensor network in which every connection between two factors is multiplied by a learnable vector $r_{k,l}^{(\\mathrm{in})}$. These rank-evolving vectors act as soft gates: an entry pushed to zero by the $\\ell^1$ penalty removes that component or edge, so the remaining nonzeros define the effective rank and the surviving structure. The mechanism does the work of replacing manual rank selection with a sparsity prior, and the fully-connected inner network supplies the flexibility to represent all-mode correlations that tensor-train or tensor-ring decompositions miss.","core_discovery":"The central discovery, on the paper's own terms, is that pairing a variable-separable outer representation with a fully-connected inner tensor network that carries per-edge learnable gate vectors makes rank selection an emergent property of optimization rather than a user decision. In HRE, each mode of the outer expansion is weighted by a vector $r^{(\\mathrm{out})}$ and each connection between inner factors is weighted by a vector $r^{(\\mathrm{in})}$; under $\\ell^1$ regularization these weights drive unneeded degrees of freedom to zero. The authors argue that after training the number of nonzero entries directly gives the suitable rank, and a pruned edge corresponds to a removed correlation, so the representation self-selects both its size and its topology. The numerical comparisons on five PDE benchmarks, supported by ablations on the 5D Poisson equation, are used to show that this adaptive structure outperforms fixed-topology tensor-based PINNs.","pith_inferences":["(Inference) The automatic rank read-out likely depends on the optimizer driving weights to precisely zero; with plain gradient-based training and an $\\ell^1$ penalty, exact zeros are rare, so a practical implementer will probably need a threshold or rounding step, which is a mild extra choice the paper does not discuss.","(Inference) The gating idea is essentially a continuous architecture search over tensor-network topologies, so it could transfer to operator learning: making the trunk and branch networks of a separable DeepONet carry the same sparse rank-evolving weights might yield adaptive operator architectures.","(Inference) The Navier-Stokes test uses one random initial vorticity field; repeating the comparison over many random fields and seeds would clarify whether the advantage reflects a universally better low-rank representation of turbulent correlations or a favorable initialization.","(Inference) Since the outer univariate networks are independent per coordinate, the same sparsity mechanism could be extended to prune whole univariate branches, shrinking the actual number of network evaluations and reducing inference cost beyond rank reduction."],"forward_implications":["If HRE-PINN's results hold, tensor-based PINNs no longer need a user-chosen rank; setting the two sparsity weights $\\lambda$ and $\\mu$ is enough, and the network discovers its own capacity.","The same hierarchy can be dropped into any PDE solver that represents the solution as a sum of separable functions, giving automatic structure selection without changing the physics-informed loss.","The ablation results indicate that the outer layer and the inner structure contribute complementary gains, so the representation should degrade gracefully when one layer is removed and improve further on problems where both the dimension count and the cross-mode coupling are large.","Because the inner network is fully connected, HRE-PINN should be able to represent solutions whose correlations are global rather than chain-like, which is where TT-PINN and TR-PINN are weakest."],"supporting_citations":[{"why":"Separable PINNs, the main CP-style baseline HRE-PINN compares against; supplies the variable-separation idea HRE extends.","marker":"[37]"},{"why":"Functional tensor decompositions for PINNs; provides the Tucker, TT, and TR baselines and the function-Tucker formulation.","marker":"[38]"},{"why":"Tensor-train decomposition; defines the TT-PINN baseline.","marker":"[44]"},{"why":"Fully-connected tensor network decomposition; the inner network structure that HRE customizes with rank-evolving vectors.","marker":"[46]"},{"why":"SVDinsTN, the structure-search method that motivates replacing fixed tensor-network topology with an evolved one.","marker":"[48]"},{"why":"Low-rank tensor function representation, the source of the function-Tucker outer decomposition.","marker":"[53]"},{"why":"The optimizer used to jointly train all parameters in the HRE-PINN framework.","marker":"[54]"},{"why":"The high-resolution flow solver used to generate the reference solution for the Navier-Stokes benchmark.","marker":"[55]"}],"fun_headline_variants":["HRE-PINNs auto-select tensor ranks, beating fixed-rank rivals","L1 gates evolve tensor ranks during training, improving PINN accuracy","Hierarchical rank-evolving PINNs adaptively find true model size","Self-tuning tensor ranks via hierarchical PINN evolution"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the $\\ell^1$ penalty together with the chosen optimizer makes redundant entries of the rank-evolving vectors exactly zero, so that the ranks and topology can be read off directly without any thresholding rule; if entries end up merely small, rank selection still depends on a manual cutoff.","fun_headline_variants_meta":{"raw":{"variants":["HRE-PINNs auto-select tensor ranks, beating fixed-rank rivals","L1 gates evolve tensor ranks during training, improving PINN accuracy","Hierarchical rank-evolving PINNs adaptively find true model size","Self-tuning tensor ranks via hierarchical PINN evolution"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.0003,"raw_usage":{"total_tokens":1756,"prompt_tokens":992,"completion_tokens":764,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":608,"completion_tokens_details":{"reasoning_tokens":693}},"tokens_in":608,"tokens_out":764,"duration_ms":8448,"temperature":1.0,"reasoning_tokens":693,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T16:39:17.229708+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Train HRE-PINN on the 5D Poisson problem and inspect the trained values of $r^{(\\mathrm{in})}$ and $r^{(\\mathrm{out})}$: count how many entries are exactly zero in floating point, and then vary the sparsity weights $\\lambda$ and $\\mu$ slightly and see whether the reported rank (the number of nonzero entries) changes discontinuously or drifts. If no exact zeros appear without a thresholding step, or if the revealed rank is highly sensitive to the penalty weight, the automatic rank determination claimed in the paper fails.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Separable PINNs, the main CP-style baseline HRE-PINN compares against; supplies the variable-separation idea HRE extends."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Functional tensor decompositions for PINNs; provides the Tucker, TT, and TR baselines and the function-Tucker formulation."},{"cited_title":"Zheng, T.-Z","cited_arxiv_id":null,"evidence_quote":"Fully-connected tensor network decomposition; the inner network structure that HRE customizes with rank-evolving vectors."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The optimizer used to jointly train all parameters in the HRE-PINN framework."}],"review_version":1}