{"id":"24f01e3e-af16-4b49-a7e0-34330f8bf0ed","arxiv_id":"2412.05714","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"This paper proposes FENNI-PGD, a hybrid surrogate method that combines finite element neural network interpolation with proper generalized decomposition, demonstrated on 1D and 2D linear and nonlinear elasticity benchmarks.","lead":"This paper builds a fast surrogate model for engineering simulations by combining two existing tricks: splitting the answer into simple building blocks and using a special neural network tied to finite-element shapes. The result is an interpretable model that can adapt its mesh while training, demonstrated on small elasticity problems.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"SVK nonlinear surrogate accuracy is never quantified: Section 4.4 reports only loss decay and displacement fields, with no error comparison against a full-order FEM solution, leaving the paper's central nonlinear claim unsupported.","rationale":"The reader's weakest_assumption focused on the separable-loss restriction and the heuristic stagnation criterion. Both are legitimate scope limitations, but the paper explicitly acknowledges the separable-loss restriction in Section 5, and the stagnation criterion is a heuristic that still produced low errors in the linear benchmark. The most load-bearing gap for the specific claim under test is the complete absence of quantitative validation for the nonlinear SVK case. This is a direct threat to the headline contribution ('non-linear surrogate modelling' in the title and abstract) because no error metric distinguishes a converged local minimum from an accurate solution. The reader's rationale did mention that 'the nonlinear case lacks error metrics,' so there is partial agreement, but the primary weakest_assumption was stated differently. My recommended verdict remains CONDITIONAL: the linear proof-of-concept with open code and small errors is credible, but the nonlinear claim needs a FEM error comparison before acceptance of the broader statement. Since the reader already assigned CONDITIONAL for overlapping reasons, I do not change the verdict; I sharpen the specific condition that must be met.","tokens_in":19998,"tokens_out":5155,"duration_ms":56118,"concrete_test":"Re-run the 2D SVK example of Section 4.4 and compare the FENNI-PGD displacement field against a reference FEM solution (e.g., FEniCS or Code_Aster) for the three parameter configurations shown in Figure 13. Compute the relative L2 error using the same norm as Table 2 (η_FEM = ||u_ROM - u_FEM|| / ||u_FEM||) and report the values. If any configuration exceeds a threshold of, say, 5%, the central claim of accurate nonlinear surrogate modeling fails; if all are below 1%, the claim is supported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's strongest claim is that FENNI-PGD produces accurate real-time surrogate models for both linear and Saint Venant-Kirchhoff (SVK) elasticity. The linear case is backed by Table 2, which reports relative L2 errors below 1% against FEM solutions. The nonlinear case, however, is validated only through loss-decay curves (Figure 12) and qualitative displacement plots (Figure 13). No error metric is computed for the SVK example, and no comparison is made to a full-order finite element solution. This matters because the SVK potential energy (Eq. 22) is a non-convex quartic functional of the displacement; gradient-based minimization over the tensor-product manifold can converge to a spurious local minimum. A monotonically decreasing training loss and a stagnating loss-decay criterion (Eq. 16) do not, by themselves, certify that the obtained low-rank solution is close to the true equilibrium solution. The conclusion asserts that 'the surrogate model has been compared to a finite element solution and shows small relative errors,' but this statement is only substantiated for the linear elasticity benchmark. Consequently, the central claim of 'non-linear surrogate modelling' is not yet empirically supported; the SVK experiment is a demonstration that the optimizer can reduce a separated-form loss, not that the resulting surrogate is accurate.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes FENNI-PGD, a hybrid reduced-order modelling method in which the modes of a Proper Generalised Decomposition are represented by FENNI networks and trained by minimising the potential energy averaged over the parameter space with Adam and L-BFGS optimisers. Modes are added greedily using a loss-stagnation criterion, the spatial mesh can be refined through a multigrid transfer-learning strategy, and the same framework is applied to linear elasticity and Saint Venant-Kirchhoff (SVK) elasticity. The paper reports a 1D analytical benchmark, a 2D two-parameter linear elasticity benchmark with relative L2 errors below 1%, and a 2D nonlinear SVK example validated only by loss-decay curves and qualitative displacement plots. The central claims are that the method builds a surrogate based on a tensor decomposition automatically on the fly, adapts the spatial and parametric discretisation during training, and provides a versatile framework for linear and nonlinear mechanics.","tokens_in":20255,"tokens_out":5487,"duration_ms":51006,"significance":"If the claims hold, the method is a useful contribution to reduced-order modelling: it combines the interpretability of finite element shape functions with automatic differentiation, allowing a PGD-like decomposition to be obtained without problem-specific solvers. The open-source implementation and interactive demos are concrete strengths that support reproducibility. The linear 2D benchmark gives quantitative evidence (Table 2) that the surrogate can approximate finite element solutions well, and the 1D analytical example shows that the method can recover an exact one-mode decomposition. The validation is not circular, since it compares against analytical and FEM reference solutions. However, the evidence is incomplete: no train/test split is described for the linear benchmark, the nonlinear SVK example has no quantitative error measure, and the greedy convergence criterion is heuristic. These issues are fixable, but they must be addressed before the general claims of accurate nonlinear surrogate modelling are accepted.","major_comments":[{"comment":"The central claim of non-linear surrogate modelling is not empirically supported. Section 4.4 reports only loss decay (Figure 12) and qualitative displacement fields (Figure 13) for the Saint Venant-Kirchhoff example, with no relative-error metric against a full-order FEM solution. The final sentence of Section 6 states that the surrogate model 'has been compared to a finite element solution and shows small relative errors,' but that comparison appears only for the linear case in Table 2. Because the SVK potential (Eq. 22) is non-convex, a decreasing training loss does not certify proximity to the equilibrium solution. Please add quantitative error evaluations, such as relative L2 errors and error maps at held-out parameter configurations, for the SVK surrogate, or restrict the nonlinear claims accordingly.","section":"Section 4.4, Figures 12-13, and Section 6"},{"comment":"No train/test split is described. Table 2 evaluates eight parameter pairs (E, θ), but the text does not state whether these pairs are part of the training sample, held out during training, or chosen after the model converged. Without an explicit held-out set and a description of the parametric quadrature used in the loss (Eq. 15), the sub-1% relative errors may be in-sample and do not establish generalisation. Please report the training/validation split, the sampling rule for the parameter space, and errors on genuinely unseen parameter points.","section":"Section 4.2.3, Table 2"},{"comment":"The stopping and mode-addition criterion is heuristic. Eq. (16) defines stagnation by the relative loss decrease ĴL = 2(L_{n-1} − L_n)/(L_{n-1} + L_n) ≤ η_c, and Algorithm 1 adds a new mode after a fixed number of stagnating epochs. However, no argument or numerical experiment links loss stagnation to the accuracy of the reduced-order solution, and the hyperparameters max_stgn, new_mode_threshold, min_useful, and η_c are free parameters. Please validate the criterion, for example by comparing the ROM error at the stopping point with errors obtained from continued training or from an independent residual-based estimator, or replace it with a certified stopping rule.","section":"Section 3.2.1, Eq. (16), and Algorithm 1"},{"comment":"The scope of the method is narrower than the abstract's general claim. Section 5 explicitly states that 'This paper only focuses on losses that can be written without building the full solution,' and notes that for non-separable losses the cost grows exponentially with the number of parameters. Since the abstract claims the method 'addresses the curse of dimensionality' for parametric problems in general, please qualify the claim: for separable losses the complexity is linear in the number of parameters, but for general non-linear losses the framework as presented does not overcome the curse. Adding a brief complexity analysis for the separable case would also help substantiate the efficiency statement.","section":"Section 5, 'Overcoming the curse of dimensionality'"}],"minor_comments":[{"comment":"The text says the second half of the bar is [L/1, L]; this appears to be a typo for [L/2, L].","section":"Section 4.1.1"},{"comment":"The phrase 'utilising the using the Multiplication block' contains a duplicated article and should be rephrased.","section":"Section 3.1"},{"comment":"The phrase 'the loss function to to be minimised' has a duplicated 'to' and should be corrected.","section":"Section 3.2"},{"comment":"The claim that '80% of the training is done on the coarse mesh' should specify whether this refers to epochs, wall-clock time, or computational cost; otherwise the statement is ambiguous.","section":"Section 3.2.3, Figure 11"},{"comment":"The comparison between sequential and simultaneous training would be more informative if it reported solution errors against the reference solution, not only loss values and mode counts.","section":"Section 4.2.2"},{"comment":"The index notation in the tensorised loss expressions, such as λ^{(0)}_{mp} and det(J)_{em}, is not defined; please specify the summation ranges and the meaning of the superscripts.","section":"Appendix A, Eq. (27) and (32)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript depends on the companion paper Part I (submitted) for the FENNI interpolation properties; if Part I is not yet available to readers, the claims here are hard to evaluate independently. Since the code and demos are publicly available, asking for a held-out test set and a quantitative nonlinear error table is a straightforward and high-value revision. The missing nonlinear error and the lack of a train/test split are the main barriers to acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper shows something genuinely useful: you can build a PGD tensor decomposition by training FENNI modes greedily with backprop, retraining all modes after each addition, and using multigrid to transfer the coarse solution to finer meshes. The 1D analytical example and the 2D linear elasticity comparison give real support for the working principle, with sub-1% relative errors against FEM. The code and interactive demos are openly available, which is more than many papers in this area do. If you work on reduced-order modelling or digital-twin workflows, this is worth knowing about.\n\nThe main soft spot is exactly where the stress-test note lands. Section 4.4 reports only loss decay and qualitative displacement plots for the Saint Venant–Kirchhoff case. No error metric is computed against a full-order FEM solution, and the conclusion's claim that \"the surrogate model has been compared to a finite element solution and shows small relative errors\" is only substantiated for the linear benchmark. Given that the SVK potential is non-convex, a decreasing training loss does not certify that the low-rank solution is close to the true equilibrium. This is a load-bearing gap for a paper titled \"non-linear surrogate modelling.\"\n\nOther soft spots are real but smaller. Table 2 does not describe a train/test split, so the reported errors may be in-sample. The convergence criterion in Eq. (16) is heuristic, with no independent error estimator. There are no baselines against standard PGD or POD-NN methods, which makes the efficiency claims hard to evaluate. The authors do honestly acknowledge in Section 5 that the method only applies to losses expressible in separated form without assembling the full solution, which undercuts the abstract's broad \"curse of dimensionality\" phrasing but is a fair limitation statement.\n\nSome reliance on unpublished Part I is a practical concern for a standalone referee, but the interpolation properties are not the core of this paper. The core—that FENNI modes can be trained greedily into a PGD surrogate—holds up for the linear case. The nonlinear case is a demonstration that the optimizer can reduce the loss, not yet evidence of accuracy.\n\nA serious referee should see this, but with clear expectations: quantify the SVK error, introduce a genuine train/test split, add at least one baseline, and temper the conclusion. The reader's conditional verdict is about right.","headline":"A credible proof-of-concept for training PGD modes with FENNI via backprop, but the paper's central nonlinear accuracy claim is not yet backed by numbers.","tokens_in":20812,"tokens_out":1444,"would_cite":false,"duration_ms":17354,"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":"This paper claims that a parametric mechanics surrogate can be assembled on the fly by training each factor of a PGD tensor decomposition as a sparse finite-element neural network, with mode count and meshes chosen automatically during…","keywords":["proper generalised decomposition","tensor decomposition","finite element neural network interpolation","parametric reduced-order modelling","linear elasticity","Saint Venant-Kirchhoff elasticity","transfer learning","adaptive mesh"],"falsifier":"One decisive test is to run the same FENNI-PGD procedure on a problem whose loss cannot be written in separated form, such as a compressible hyperelastic model with a coupling term mixing deformation gradient and parameter fields, and measure how wall-clock time per epoch scales as the number of parameters grows from 2 to 5, since exponential rather than roughly linear scaling would contradict the curse-of-dimensionality claim; alternatively, on the paper's own 2D example, check whether the loss-stagnation stopping point coincides with low relative L2 error against a fine finite element solution.","tokens_in":19785,"feed_emoji":"⚙️","tokens_out":8813,"duration_ms":78987,"temperature":0.7,"pith_summary":"The paper proposes FENNI-PGD, a reduced-order surrogate method that builds a Proper Generalised Decomposition (PGD) of a parametrised mechanics problem using the Finite Element Neural Network Interpolation framework from Part I. Instead of collecting high-fidelity snapshots, it trains the space and parameter modes of a tensor decomposition directly, so the reduced-order basis is available as soon as training ends. The claim is that this can be done automatically: the number of modes is decided greedily by loss stagnation, the spatial and parametric meshes are refined during training, and changing the physics only means changing the loss function. If true, this removes most of the manual choices that currently make PGD solvers problem-dependent, and makes real-time parametric elasticity simulations, such as soft-tissue digital twins, much more accessible. The paper demonstrates the idea on 1D and 2D linear elasticity and a Saint Venant-Kirchhoff finite-strain example, with relative errors below one percent on the tested parameter configurations.","feed_headline":"Auto-trained surrogate solves parametrized elasticity in real time","feed_subtitle":"Each tensor mode is a sparse finite-element neural network, so mode count and meshes adapt during training","key_machinery":"The central object is the FENNI-PGD tensor decomposition $u(x,\\{\\mu_i\\}) = \\sum_{i=1}^m u_i(x) \\prod_{j=1}^{\\beta} \\lambda_i^j(\\mu_j)$, in which each mode factor is a FENNI module, i.e., a neural network constrained to reproduce finite element shape functions whose parameters are nodal values and nodal coordinates. Three mechanisms drive the argument: a greedy loop that appends a mode whenever the relative loss decrease falls below a threshold in Eq. (16); two-stage optimisation combining Adam for exploration and L-BFGS for fine-tuning; and separated loss expressions computed by tensor contraction that avoid forming the full solution. The multigrid training strategy completes the machinery by transferring a converged coarse decomposition to finer meshes, so mesh refinement and convergence studies happen as part of training.","core_discovery":"The central claim is that the FENNI framework can serve as an elementary building block for generating PGD modes: each factor in the tensor decomposition is a sparse neural network whose trainable parameters are finite element nodal values and coordinates, and the tensor product structure is assembled inside the network. The decomposition is built greedily during training, but the authors find that retraining all modes together after each addition yields a lower-rank decomposition than the classical sequential strategy. Because the potential-energy loss is written in separated form for linear elasticity and Saint Venant-Kirchhoff elasticity, the full-order solution is never assembled, and the training is carried out by backpropagation with Adam followed by L-BFGS. On the numerical examples, the method selects the number of modes on its own, transfers coarse modes to finer meshes by interpolation, and produces surrogates whose errors against reference finite element solutions stay below one percent.","pith_inferences":["Beyond the paper: the separable-loss condition is the real boundary of the method; any constitutive model whose energy is polynomial in modes could be implemented without full assembly, but non-polynomial or history-dependent behaviour would not.","Beyond the paper: because all modes are retrained together, the same simultaneous-update idea could be used as a post-processing step in classical PGD solvers to improve rank optimality without changing the greedy enrichment.","Beyond the paper: the paper's own suggestion of coupling the decomposition with an auto-encoder latent space points to a natural test on a problem with poor low-rank approximability, where the mode count rather than the mesh size would reveal the method's practical limits.","Beyond the paper: the loss-stagnation stopping rule has no certified link to solution error; a practical extension would replace or supplement it with a residual-based error estimator to make automatic mode selection trustworthy in safety-critical applications."],"forward_implications":["Surrogate models for parametric elasticity can be constructed without any full-order solves or snapshot selection, so the offline phase reduces to training a single network architecture.","The number of modes is selected automatically, replacing the manual rank selection and convergence studies of classical PGD.","Changing from linear to finite-strain elasticity only requires swapping the energy expression in the loss, so the same code path handles different physics.","Because modes are interpretable and transferable, a model trained on a coarse mesh can be refined and reused, cutting the cost of fine-mesh training.","Once trained, the surrogate is evaluated by summing products of modes, so new parameter values are obtained in real time without solving the PDE."],"supporting_citations":[{"why":"Supplies the FENNI framework, including its interpretable adaptive-mesh training, whose mode networks are reused here.","marker":"Škardová et al., 2024"},{"why":"Introduces the HiDeNN construction that embeds finite element shape functions and the Multiplication block inside the network.","marker":"Zhang et al., 2021"},{"why":"Proposes the HiDeNN tensor decomposition that FENNI-PGD turns into a greedy, automatically sized surrogate.","marker":"Zhang et al., 2022"},{"why":"Provides the Proper Generalised Decomposition concept and its a priori reduced-order modelling rationale.","marker":"Chinesta et al., 2011"},{"why":"Documents the problem-dependent difficulty of standard PGD minimisation that the backpropagation-based training aims to bypass.","marker":"Nouy, 2010"},{"why":"Is the open-source implementation used to produce the paper's numerical results.","marker":"Daby-Seesaram et al., 2024b"},{"why":"Supplies the Adam optimiser used in the first exploration stage of the two-stage training.","marker":"Kingma & Ba, 2014"}],"fun_headline_variants":["Joint retraining yields lower-rank PGD surrogates","Sparse neural nets assemble PGD modes adaptively","Self-adapting surrogate for parametrized mechanics","FENNI-PGD: transferable modes, real-time training"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the loss can be evaluated in separated form—as sums of products of space-only and parameter-only terms—without ever assembling the full solution, with a secondary reliance on a heuristic stagnation criterion that is not proven to track surrogate accuracy.","fun_headline_variants_meta":{"raw":{"variants":["Joint retraining yields lower-rank PGD surrogates","Sparse neural nets assemble PGD modes adaptively","Self-adapting surrogate for parametrized mechanics","FENNI-PGD: transferable modes, real-time training"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000285,"raw_usage":{"total_tokens":1683,"prompt_tokens":952,"completion_tokens":731,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":568,"completion_tokens_details":{"reasoning_tokens":664}},"tokens_in":568,"tokens_out":731,"duration_ms":8096,"temperature":1.0,"reasoning_tokens":664,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T20:25:26.642228+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"One decisive test is to run the same FENNI-PGD procedure on a problem whose loss cannot be written in separated form, such as a compressible hyperelastic model with a coupling term mixing deformation gradient and parameter fields, and measure how wall-clock time per epoch scales as the number of parameters grows from 2 to 5, since exponential rather than roughly linear scaling would contradict the curse-of-dimensionality claim; alternatively, on the paper's own 2D example, check whether the loss-stagnation stopping point coincides with low relative L2 error against a fine finite element solution.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Documents the problem-dependent difficulty of standard PGD minimisation that the backpropagation-based training aims to bypass."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Adam optimiser used in the first exploration stage of the two-stage training."}],"review_version":1}