{"id":"3e26bf5e-4a17-43dc-b894-76eab277dae9","arxiv_id":"2501.16370","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":3.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"A residual-connection neural network (RISN) solves integral and integro-differential equations with lower mean absolute error than PINN, A-PINN, and SA-PINN on most of 20 benchmark problems.","lead":"This paper adds skip connections to a physics-informed neural network so it can solve integral equations, like equations with memory or oscillations, more accurately. The authors report lower errors than three existing PINN variants on 20 test problems, though the results are not always consistent.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Residual-connection attribution is confounded: quadrature/operational-matrix machinery is credited to PINNIES but no PINNIES or identical-numerics baseline is compared.","rationale":"The key load-bearing claim is causal: residual connections improve accuracy and stability. The paper's own methodology section undermines the causal isolation: Gaussian quadrature and fractional operational matrices were introduced by PINNIES, and RISN uses them. Since the numerical integration scheme can dominate error—the paper's Abel discussion concedes this—the comparison must hold integration fixed across models. The described protocol does not guarantee that; it only fixes architecture, optimizer, and loss terms. Without a PINNIES baseline or code, a reader cannot tell whether residual connections or the numerical layer explains the MAE reductions. This is why I consider this the weakest point, not the quadrature error bound per se. The reader's weakest assumption (fixed 50-point quadrature accuracy below network error) is adjacent but different; even a very accurate quadrature would not save the claim if the baselines did not use it. I also note the paper's 'consistently outperforms' wording is contradicted by its own Table 3 (Volterra system, second equation: PINN 1.98e-5 vs RISN 1.18e-4) and Table 6 Problem 5 tie, so the overclaim needs softening. The sensitivity analysis and explicit hardware/optimizer reporting are useful and support training stability, but they do not address attribution. Verdict remains CONDITIONAL: the method is plausible, but the central advantage claim needs the PINNIES/identical-numerics control, error bars, and code release.","tokens_in":19594,"tokens_out":6963,"duration_ms":69340,"concrete_test":"Run a controlled comparison on the 20 Table 6 problems: implement PINN, A-PINN, SA-PINN, and PINNIES with the exact same 50-point Gauss-Legendre quadrature and the same fractional operational matrix routines as RISN, with PINNIES differing only by the absence of residual connections. If PINNIES (or a residual-free PINN with identical operators) reaches the same MAEs as RISN, the reported gains are due to the numerical integration layer, not the residual architecture. Separately rerun Problems 5, 6, and 20 with 100+ quadrature nodes and an Abel-adapted rule; if the Abel tie or the fractional MAE changes materially, the 'consistently outperforms' headline needs revision.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim is that RISN's residual connections yield consistently lower MAE. The evidence does not isolate that mechanism. Section 3.2 explicitly credits PINNIES [39] with first integrating Gaussian quadrature and fractional operational matrices into a PINN framework, and says RISN's advantage combines these techniques with residual connections. The experimental protocol (Section 4) states that all models share network depth/width, optimizer, and loss components, but never states that the PINN, A-PINN, and SA-PINN baselines evaluate integral and fractional operators with the same 50-point Gauss-Legendre quadrature and the same fractional operational matrix. If the baselines used cruder or different discretizations, the improvements in Tables 1-6—including the 'failed to solve' entries for Abel, Volterra, and fractional problems—would be caused by the numerical operators, not by the skip connections. The natural control that separates these factors, a residual-free PINNIES baseline with identical numerical operators, is absent, and no code is provided to infer the implementation. The paper's own admission that Abel accuracy depends on quadrature nodes or a different rule (Section 4.1.1) shows the numerical layer can dominate error, so the confounding is not hypothetical.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes RISN, a fully connected neural network with residual (skip) connections for solving integral and integro-differential equations, including one- and multi-dimensional problems, systems, fractional equations, and a Helmholtz-type oscillatory-kernel problem. Integral terms are evaluated with Gauss-Legendre quadrature and fractional derivatives with operational matrices, and training minimizes a weighted MSE of the equation residual plus initial/boundary/data terms. The paper compares RISN with PINN, A-PINN, and SA-PINN on 20 benchmark problems, reporting MAE values, and claims that residual connections stabilize gradient flow and yield consistently lower MAE than all baselines.","tokens_in":19910,"tokens_out":6664,"duration_ms":63737,"significance":"If substantiated, a robust architecture covering this breadth of integral and integro-differential equations would be a useful contribution to scientific machine learning. The paper has real strengths: the benchmark suite is broad, MAE is measured against independent exact solutions for most problems, the experimental section states that all methods share the same network architecture and optimizer, and the sensitivity analysis addresses depth and learning rate. However, the central quantitative claim is not supported by the paper's own data, and the attribution of the gains to residual connections is confounded by the numerical-operator implementation. The result is potentially salvageable, but only after a controlled comparison and a substantially revised set of claims.","major_comments":[{"comment":"The abstract and Sections 5-6 claim that RISN 'consistently outperforms' baselines and 'achieves the lowest MAE across almost all problems,' but Table 6 contradicts this. Problem 15 (first-kind Volterra ordinary integro-differential equation) reports A-PINN MAE 1.70e-5 versus RISN MAE 5.38e-4, so A-PINN is roughly 30 times more accurate. In Problem 10, second component, PINN achieves 1.98e-5 versus RISN 1.18e-4, and in Problem 5 the values are an exact tie at 3.27e-3. Table 3 already concedes one Volterra case where PINN outperforms RISN. The overarching performance claim must be replaced by a per-problem, quantified statement, and the distribution of wins/losses/tie should be reported explicitly.","section":"Section 4.3, Table 6"},{"comment":"The claimed causal role of residual connections is not isolated. Section 3.2 credits PINNIES [39] with first integrating Gaussian quadrature and fractional operational matrices into a PINN framework and states that these techniques 'do not significantly differ from their original implementation.' The experimental protocol in Section 4 guarantees identical network architecture and optimizer settings across methods, but it never states that the PINN, A-PINN, and SA-PINN baselines evaluate integral and fractional operators with the same 50-point Gauss-Legendre rule and the same fractional operational matrices. A residual-free PINNIES baseline with identical numerical operators is the natural control and is absent. Because Section 4.1.1 itself notes that Abel-type accuracy depends on the number of quadrature nodes or the choice of quadrature rule, the numerical layer can dominate the error; without this control, the improvements in Tables 1-6 cannot be attributed to residual connections.","section":"Sections 3.2 and 4"},{"comment":"The Sobolev-space argument is not a proof of an advantage for RISN. It assumes a known approximation g with small W^{k,p} error and asserts that the residual r is smoother, but no such g is constructed in the RISN pipeline, no approximation rate with explicit dependence on width and depth is stated, and the cited results [58-60] are universal-approximation statements rather than quantitative Sobolev bounds for the specific residual decomposition used here. The 'implication' therefore does not follow from the cited mathematics. This subsection should either be removed or replaced with a rigorous statement; in its current form it overstates the theoretical grounding.","section":"Section 3.3"},{"comment":"All experimental comparisons report single MAE values with no number of seeds, repeated runs, error bars, or statistical tests. Since L-BFGS training is initialization-dependent, claims of 'robustness,' 'stability,' and 'consistently' lower error are not supported by the evidence. The absence of code or data also prevents checking whether the baseline implementations share the numerical operators. Reporting at least a small number of independent runs with dispersion measures is necessary before the comparative conclusions can be drawn.","section":"Tables 1-6 and Figure 3"}],"minor_comments":[{"comment":"The loss definition repeats lambda_IC; the second coefficient should presumably be lambda_BC.","section":"Equation (3)"},{"comment":"The roles of F, D, and I are described inconsistently: one sentence says F(u), D(u), I(u) correspond to differential, integral, and source terms, while the following text says D(u) is the differential operator. Clarify the notation.","section":"Section 3.1 and Figure 1"},{"comment":"The Helmholtz problem uses a Neumann-series reference solution on a discretization grid of 0.001, but no validation of the reference accuracy is given. Without an error estimate for the reference, statements such as 'MAE of 1.46e-3' versus '7.51e-3' assume that the reference is much more accurate than both values.","section":"Section 4.4"},{"comment":"The paper lists 'integration of advanced numerical techniques' as a contribution, but Section 3.2 states that the Gaussian quadrature and fractional operational matrices were first integrated into a PINN framework by PINNIES [39] and 'do not significantly differ from their original implementation.' The novelty framing should be revised to focus on the residual architecture and to acknowledge that the numerical techniques are inherited.","section":"Introduction and Section 3.2"},{"comment":"The bullet 'In 18 out of 20 problems, RISN delivers either the best or near-best accuracy' is unverifiable because 'near-best' is not defined. Provide an exact ranking criterion and a complete win/loss/tie table.","section":"Section 4.3.1"},{"comment":"Several table headings contain typos such as 'T able' and 'F ractional'; these should be corrected.","section":"Tables 1-6"}],"recommendation":"major_revision","confidential_remarks":"The paper's main claimed novelty over PINNIES is the residual architecture, but the authors are affiliated with the PINNIES work and no PINNIES baseline is included; this is a serious gap because Section 3.2 explicitly says the numerical machinery is unchanged. I would ask the authors to add a residual-free PINNIES control and to rerun the comparison with fixed seeds and repeated trials. The contradiction in Table 6 (Problem 15) is the key data point that must be addressed; it is fixable by narrowing the claim, but the current abstract and conclusions overstate the results."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a plausible incremental methods paper with a genuinely useful benchmark suite, but the headline accuracy claim is false on its own data and the experiments do not isolate the mechanism the title points to. I would send it to review, but only with major revision.\n\nWhat's actually new: the combination of residual skip connections with the PINNIES-style quadrature and fractional operational matrix machinery, tested on a broad set of 20 integral and integro-differential problems—1D, multi-D, systems, fractional, and an oscillatory Helmholtz kernel. The tables are easy to read, and the authors credit PINNIES for the numerical operator components rather than pretending those are new. The sensitivity analysis on depth and learning rate gives qualitative evidence that residual connections stabilize training.\n\nThe soft spots are real. The phrase \"consistently achieves the lowest MAE across almost all problems\" is contradicted in Table 6: in Problem 15, A-PINN gets 1.70e-5 while RISN gets 5.38e-4, two orders of magnitude worse; in Problem 10, PINN beats RISN on the second function; Problem 5 is a tie. The honest claim is \"often better, sometimes worse.\"\n\nMore important, the residual connection is not isolated as the cause. Section 3.2 explicitly says the Gaussian quadrature and fractional operational matrices are taken unchanged from PINNIES. The paper compares against PINN, A-PINN, and SA-PINN, but never against PINNIES or a residual-free network using the exact same numerical quadrature and fractional matrices. Without that control, the improvements could come from the discretization rather than the skip connections. The paper doesn't even state that the baselines used the same 50-point Gauss-Legendre rule. The stress-test concern is legitimate: this is a confound, not a hypothetical. The Abel comment in Section 4.1.1 confirms the numerical layer can dominate—there both methods tie at 3.27e-3.\n\nAlso: no error bars, no seeds, no statistical tests, and the sensitivity plots look like single runs. The Helmholtz reference is a numerically constructed Neumann series, so it is not an independent check. The Sobolev section is a heuristic motivation, not a theorem, and should be labeled as such.\n\nNone of this kills the paper. The benchmark suite is useful, the method is plausible, and the failures are fixable. I would want a PINNIES or identical-numerics baseline, corrected claims, error bars, and a code release before accepting. For someone working on neural solvers for integral equations, this is worth reading; for a general reader, it is a standard incremental methods paper.","headline":"A plausible incremental idea—residual connections for PINNs solving integral equations—with a useful benchmark suite, but the paper's own tables contradict the 'consistently outperforms' claim and the experiments never isolate the residual mechanism from the PINNIES-style numerical operators.","tokens_in":20413,"tokens_out":2835,"would_cite":false,"duration_ms":28650,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["68T07","65R20"],"pacs":[],"model":"deepseek-v4-flash","headline":"Adding residual skip connections to a physics-informed neural network yields lower errors on almost every integral and integro-differential equation tested, including fractional, multi-dimensional, and oscillatory-kernel problems.","keywords":["residual connections","physics-informed neural networks","integral equations","integro-differential equations","fractional derivatives","Gaussian quadrature","fractional operational matrices","Helmholtz-type integral equations"],"falsifier":"Remove only the residual skip connections from RISN while keeping depth, optimizer, quadrature points, loss terms, and seeds identical, then re-run all 20 benchmarks; if the MAE does not consistently rise, the paper's central attribution of accuracy gains to residual connections is not supported.","tokens_in":19420,"feed_emoji":"🧮","tokens_out":7514,"duration_ms":73866,"temperature":0.7,"pith_summary":"RISN is a physics-informed neural network for integral and integro-differential equations that adds a residual skip connection to every hidden layer, so each layer computes $\\sigma(A_{i-1}\\theta^{(i)}+b^{(i)}) + A_{i-1}$. The paper's claim is that this single architectural change stabilizes gradient flow and lets the network train at greater depth, producing more accurate solutions than the plain PINN and than the A-PINN and SA-PINN variants across a 20-problem benchmark suite. The measured mean absolute errors are often one to two orders of magnitude lower, and on several hard fractional, system, and oscillatory-kernel problems the variants fail to converge while RISN does. If this is right, residual connections are a cheap, broadly applicable improvement for neural solvers of nonlocal equations, not a method-specific fix.","feed_headline":"Residual connections cut PINN error up to 100x on integral equations","feed_subtitle":"New RISN solver beats PINN and two advanced variants across 20 benchmarks, including fractional and oscillatory kernels.","key_machinery":"The carrying mechanism is the residual skip connection: each hidden layer is defined by $A_i = \\sigma(A_{i-1}\\theta^{(i)} + b^{(i)}) + A_{i-1}$, which gives gradients a direct path from the output back through the network and is credited with preventing vanishing gradients. Around this sit the numerical operators: Gauss–Legendre quadrature with 50 points evaluates integral terms, and fractional operational matrices evaluate Caputo fractional derivatives. The paper also argues from Sobolev-space approximation that the residual network learns a correction $r(x)=y^*(x)-g(x)$ to a coarse approximation, a smoother target than the raw solution, so the total approximation error inherits the network's bound on $r$.","core_discovery":"The central discovery claimed is that the residual connection itself—not the quadrature rule or the fractional operational matrices, which are shared with earlier PINN-based solvers—carries the accuracy gain. RISN reports the best or near-best mean absolute error in 18 of 20 benchmark problems, including a tie on the linear Abel equation, and it is the only method that converges on the nonlinear Abel problem and several first-kind systems. Across the suite the advertised improvement is up to two orders of magnitude over the baseline PINN and at least fivefold over A-PINN and SA-PINN on the hardest cases. The Helmholtz-type equation, whose solution is known only through a Neumann-series reference with grid step 0.001, is included to show that the gain persists when no exact analytical solution is available.","pith_inferences":["Because the compared methods share the same quadrature and fractional-matrix approximations, the claimed two-order gains are best read as optimization and architecture gains for a fixed discretization; changing those discretizations could shrink or enlarge the gap depending on whether the numerical operator error dominates.","A direct ablation not reported in the paper—removing only the skip connections from RISN while keeping depth, optimizer, quadrature, and loss identical—would isolate whether the residual path itself is the cause of the accuracy gain.","The Sobolev argument suggests RISN's advantage should grow when a good coarse approximation $g$ is available; testing the method with deliberately poor choices of $g$ would separate the benefit of a smoother correction target from the benefit of improved gradient flow."],"forward_implications":["Any PINN-style integral-equation solver can adopt residual connections without changing its quadrature or loss structure, making the improvement a drop-in architectural change.","Deeper networks become usable: the sensitivity analysis shows smoother loss curves and lower final MAE for RISN as depth grows from 2 to 10 hidden layers, whereas PINN's MAE spikes with depth.","The hard cases where A-PINN and SA-PINN fail—fractional integro-differential equations, strongly coupled Volterra systems, and 2D Volterra equations—are exactly where the residual architecture matters most.","For singular Abel kernels, accuracy remains limited by the quadrature, not the network, so users must increase nodes or switch quadrature to push past the reported $3.27\\times10^{-3}$ floor.","In the no-exact-solution Helmholtz problem, RISN's MAE of $1.46\\times10^{-3}$ against the Neumann-series reference indicates oscillatory kernels are not a barrier to convergence, though the reference is itself a numerical construction."],"supporting_citations":[{"why":"Defines the baseline PINN framework that RISN extends and is compared against.","marker":"[25]"},{"why":"Supplies the residual-connection idea that is the paper's central architectural innovation.","marker":"[33]"},{"why":"A-PINN, one of the two advanced variants RISN must beat; the comparison against it frames the claimed advantage.","marker":"[37]"},{"why":"SA-PINN, the adaptive-loss variant that fails on the hardest problems and anchors the robustness claim.","marker":"[38]"},{"why":"Provides the integration of Gaussian quadrature and fractional operational matrices into the PINN framework, which RISN inherits.","marker":"[39]"},{"why":"Gives the fractional integro-differential benchmark problem and its exact solution used in Example 2.","marker":"[22]"},{"why":"Supplies the numerical-analysis basis for Gaussian quadrature as the integral evaluation method.","marker":"[21]"}],"fun_headline_variants":["RISN: Residual nets solve integral equations 100x better than PINN","Residual links cut integral-solver error by 100x over PINN","New net uses residual tricks to beat PINN on hard integrals","Residual connections give 100x error drop for tricky integral equations","Solver with residual links beats PINN, A-PINN, SA-PINN on 20 tests"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The comparison assumes that the fixed 50-point Gauss–Legendre quadrature and the fractional operational matrices approximate every integral and fractional operator with error far below the network error, so that the reported MAE differences reflect the residual architecture rather than the numerical discretization.","fun_headline_variants_meta":{"raw":{"variants":["RISN: Residual nets solve integral equations 100x better than PINN","Residual links cut integral-solver error by 100x over PINN","New net uses residual tricks to beat PINN on hard integrals","Residual connections give 100x error drop for tricky integral equations","Solver with residual links beats PINN, A-PINN, SA-PINN on 20 tests"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000864,"raw_usage":{"total_tokens":3731,"prompt_tokens":913,"completion_tokens":2818,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":529,"completion_tokens_details":{"reasoning_tokens":2715}},"tokens_in":529,"tokens_out":2818,"duration_ms":19744,"temperature":1.0,"reasoning_tokens":2715,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T16:22:23.400855+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Remove only the residual skip connections from RISN while keeping depth, optimizer, quadrature points, loss terms, and seeds identical, then re-run all 20 benchmarks; if the MAE does not consistently rise, the paper's central attribution of accuracy gains to residual connections is not supported.","supporting_citations":[{"cited_title":"In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp","cited_arxiv_id":null,"evidence_quote":"Supplies the residual-connection idea that is the paper's central architectural innovation."},{"cited_title":"Journal of Computational Physics462, 111260 (2022)","cited_arxiv_id":null,"evidence_quote":"A-PINN, one of the two advanced variants RISN must beat; the comparison against it frames the claimed advantage."},{"cited_title":"Journal of Computational Physics474, 111722 (2023)","cited_arxiv_id":null,"evidence_quote":"SA-PINN, the adaptive-loss variant that fails on the hardest problems and anchors the robustness claim."},{"cited_title":"PINNIES: An Efficient Physics-Informed Neural Network Framework to Integral Operator Problems","cited_arxiv_id":"2409.01899","evidence_quote":"Provides the integration of Gaussian quadrature and fractional operational matrices into the PINN framework, which RISN inherits."},{"cited_title":"Journal of Vibration and Control17(13), 2050– 2058 (2011)","cited_arxiv_id":null,"evidence_quote":"Gives the fractional integro-differential benchmark problem and its exact solution used in Example 2."},{"cited_title":"John wiley & sons, ??? (1991)","cited_arxiv_id":null,"evidence_quote":"Supplies the numerical-analysis basis for Gaussian quadrature as the integral evaluation method."}],"review_version":1}