{"id":"5f35c76e-8e82-42b9-8f70-fc62bf9beeb4","arxiv_id":"2507.06817","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":5,"one_line_summary":"A neural network that learns time-varying observer gains, combined with adaptive sliding mode control, estimates states of general nonlinear systems directly from sensor measurements.","lead":"This paper trains a neural network to generate time-varying observer gains and adds an adaptive sliding-mode correction term to estimate states of nonlinear systems from sensor outputs without linearizing the dynamics. The authors give a convergence proof and test the observer on chaotic, non-differentiable, and weakly observable systems.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1's decisive hypothesis is exponential decay of the NN approximation error epsilon_k; no training or architecture result supplies this, so the proof assumes a condition as strong as the advertised convergence and the central guarantee is unsupported.","rationale":"The paper's strongest claim is the exponential-convergence guarantee of Theorem 1. For that theorem to hold, the neural-network approximation error must decay exponentially in time. The authors invoke the Universal Approximation Theorem, but that theorem only ensures uniform approximation accuracy over a compact set—it says nothing about how the approximation error behaves along a trajectory, let alone that it decays exponentially. The training procedure and loss function do not target epsilon_k directly, and no width, depth, or optimization guarantee is given. Consequently, Assumption 4 plus the extra exponential condition is effectively a restatement of the desired behavior rather than a condition the method can be shown to satisfy. This is the load-bearing weak point: if the approximation error does not decay as assumed, the proof degrades to the ultimate-boundedness result of Corollary 1, which is not the advertised exponential guarantee. The reader's weakest-assumption analysis identifies exactly this issue, so I agree with that assessment. I also note the proof's Step 8 sector inequality is false globally, which is a further internal inconsistency, but the exponential-approximation assumption is the more fundamental obstacle to the central claim. The proposed concrete test—measuring epsilon_k on one of the paper's own examples—would settle whether the theorem's hypothesis is ever satisfied. Without such verification, and without code or data to reproduce the simulations, the central contribution is not supported. This confirms the reader's REJECT verdict; the paper's theoretical guarantee is conditional on an unverified, essentially-as-strong-as-the-conclusion assumption, and the proof itself contains a false inequality.","tokens_in":16384,"tokens_out":6840,"duration_ms":81380,"concrete_test":"Reproduce Example 1 (Rössler) using the paper's described training: generate y_k from Eq. (38), train the MLP with loss (10)-(13), and at each test time compute L_k from the network and L*_k by solving the discrete Riccati inequality from Step 7 (Eq. 29). Record epsilon_k = ||L_k - L*_k|| along the test trajectory and test whether epsilon_k <= epsilon0 e^{-lambda t_k} for any positive lambda and epsilon0. If no such exponential envelope fits the observed approximation error, the hypothesis of Theorem 1 is not satisfied, and the reported rapid convergence cannot be attributed to the theorem. Also report whether the actual error bound matches Corollary 1's C epsilon_max ultimate bound rather than the advertised exponential decay.","verdict_should_be":"REJECT","load_bearing_attack":"Assumption 4 (Section V) only states that the neural-network approximation error is bounded, ||epsilon_k|| <= epsilon_max. Theorem 1 then adds the much stronger condition ||epsilon_k|| <= epsilon0 e^{-lambda t_k}. This exponential-decay requirement is not derived from the Universal Approximation Theorem, from any width/depth bound, from the training loss (10)-(13), or from the optimization procedure. Universal approximation is a static statement over a compact set; it gives no pointwise-in-time decay guarantee along a trajectory. Since epsilon_k = L_k - L*_k is the discrepancy between the learned time-varying gain and the ideal stabilizing gain, requiring it to vanish exponentially is an external, unchecked condition that is as strong as the convergence the theorem promises. The Lyapunov analysis in Steps 5-10 (Eqs. 25-36) yields only ultimate boundedness when epsilon_k is merely bounded, as Corollary 1 explicitly concedes; the advertised exponential convergence to a small region comes entirely from the assumed exponential decay of epsilon_k. Thus the central claim is conditional on a hypothesis that is neither guaranteed by the method nor verified in the experiments. A separate proof defect, the false sector inequality e^T H^T tanh(H e) >= sigma ||H e||^2 in Step 8 (Eq. 30) for large ||H e||, further weakens the theorem even under this assumption.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a discrete-time nonlinear observer whose time-varying gain is produced by a multilayer perceptron trained with a loss that combines a dynamics residual, an output error, and weight regularization, together with an adaptive sliding-mode correction term. Section V states Theorem 1, which claims exponential convergence of the estimation error to a bounded region under Assumptions 1-4 and an exponential bound on the neural network approximation error. The paper reports simulations on seven examples, including chaotic, weakly observable, and non-differentiable systems, and compares the method against learning-based observers.","tokens_in":16854,"tokens_out":6448,"duration_ms":73701,"significance":"If the theoretical claim were sound, the framework would be significant: it offers an observer design that avoids linearization and coordinate transformations and uses only measured outputs for training. The empirical part covers diverse benchmarks and reports favorable error metrics relative to supervised NN, unsupervised AE, and supervised PINN baselines, which is a genuine strength of the paper. However, the central convergence proof is not valid as stated, and the key exponential-error assumption is neither derived from the training procedure nor verified in the experiments; the contribution is therefore currently an empirical observer design without a rigorous stability guarantee.","major_comments":[{"comment":"The inequality e^T H^T tanh(H e) >= sigma ||H e||^2 for some global sigma > 0 is false because tanh saturates. For any scalar z, z tanh(z) behaves like |z| as |z| tends to infinity, so z tanh(z) >= sigma z^2 cannot hold for all z with a fixed sigma > 0; the same holds componentwise for the vector expression. Consequently the lower bound on the SMC contribution used in Step 9, condition (32), does not follow, and the Lyapunov decrease attributed to the sliding-mode term is unproven.","section":"V, Step 8, Eq. (30)"},{"comment":"Equations (20)-(21) assert quadratic remainder bounds ||delta_f(e)|| <= M_f/2 ||e||^2 and ||delta_h(e)|| <= M_h/2 ||e||^2. These require f and h to be twice continuously differentiable with bounded second derivatives on a convex set containing the segment between x_k and hat(x)_k. Assumption 2 only states continuous differentiability with bounded Jacobians, and the compact set Omega is never defined; moreover Assumption 3 evaluates F_k at hat(x)_k, which is not the evaluation point supplied by the mean value theorem. The proof is therefore not supported by the stated assumptions, and the reuse of M_f and M_h with different meanings adds further ambiguity.","section":"V, Eqs. (20)-(21), Assumption 2"},{"comment":"The exponential decay assumption ||epsilon_k|| <= epsilon0 e^{-lambda t_k} is an additional hypothesis of Theorem 1, not a consequence of Assumption 4 or of the training loss (10)-(13). Universal approximation is a static statement on a compact set; it provides no pointwise-in-time decay guarantee for epsilon_k = L_k - L*_k along a trajectory. Since epsilon_k is exactly the discrepancy between the learned and the ideal stabilizing gain, assuming it decays exponentially is close to assuming the conclusion that the observer error converges. Corollary 1 shows that with only boundedness one gets ultimate boundedness, so the advertised exponential convergence rests entirely on this unverified condition.","section":"V, Theorem 1 and Assumption 4"},{"comment":"Conditions (32) and (33) contain ||e_k|| on the right-hand side, so they are state-dependent and cannot be verified a priori. In particular, (33) requires gamma to exceed a term proportional to ||e_k||, so large initial errors can violate the condition at the start of the very process whose boundedness the theorem aims to establish; no argument is given that the required inequality holds along the trajectory. This makes the stability proof circular at a load-bearing point.","section":"V, Step 9, conditions (32)-(33)"},{"comment":"The comparison principle is misstated. From Delta V <= -rho V_k + theta one obtains V_{k+1} <= (1-rho) V_k + theta, and the solution is (1-rho)^k V_0 plus theta/rho, not e^{-rho k} V_0 plus theta/(1-e^{-rho}) unless one assumes V_{k+1} <= e^{-rho} V_k + theta and 0 < rho < 1. The constant theta is never explicitly bounded, and no uniform bounds on lambda_max(P_k) are given, so inequality (36) does not follow from the preceding steps.","section":"V, Step 10, Eqs. (34)-(36)"}],"minor_comments":[{"comment":"Assumption 2 introduces a compact set Omega that is never defined; it should be X or the convex hull of X.","section":"V, Assumption 2"},{"comment":"Table I lists y = x1 for Example 1, while the Roessler system in Eq. (38) has y = x2; one of the two is a typo.","section":"VI, Example 1 and Table I"},{"comment":"The caption of Fig. 5 refers to 'system (43)', but Example 4 concerns system (41); the three-tank system is (43).","section":"VI, Example 4, Fig. 5"},{"comment":"The system in Eq. (44) is called a reverse Duffing oscillator, but the displayed equations do not match the Duffing oscillator form; please clarify the model or the label.","section":"VI, Example 7, Eq. (44)"},{"comment":"The evaluation reports single-run error metrics without confidence intervals or multiple-seed statistics; comparisons in Table III should state the variability across training runs.","section":"VI, Table III"}],"recommendation":"reject","confidential_remarks":"The paper has a substantial empirical component, but the theory section is the advertised core of the contribution. The errors in Theorem 1 are not cosmetic: the false tanh sector bound, the unproved quadratic remainder bounds, and the exponential approximation-error assumption jointly invalidate the main theorem. I would not consider this a minor revision. If the authors can either remove the stability claims and present the work as an empirical observer study, or supply a correct and genuinely proved convergence theorem with assumptions that are verified in training, a resubmission could be considered. There is also a mismatch between the title and abstract promise of a 'nearly universal' observer and the actual theoretical support provided."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nQuick take: this is a plausible engineering recipe that is overclaimed as a general theory. The paper combines a Luenberger observer with an NN-generated time-varying gain and a tanh sliding-mode term, and trains the NN with a physics-informed loss that uses only measurements, not ground-truth states. That is a useful direction, and the benchmark set (Rössler, three-tank, reverse Duffing, modified outputs) is more varied than most papers in this area. The comparison numbers in Table III are also encouraging if they hold up. Credit where due: the empirical section is honest about challenges like non-differentiable dynamics and weak observability.\n\nThe problems are in the theory and the training objective, and they are load-bearing.\n\nFirst, Theorem 1's proof breaks at Eq. (30): the claim e^T H^T tanh(H e) >= sigma ||H e||^2 is false for large ||H e|| because tanh saturates. You get linear growth, not quadratic. The sector condition needed for SMC stability simply isn't there in the global form used.\n\nSecond, Eqs. (20)-(21) invoke Taylor remainders with quadratic bounds, but Assumption 2 only assumes C^1 with bounded Jacobians. On a compact set, C^1 gives a linear remainder, not quadratic, unless second derivatives are bounded. So the Lyapunov difference bound is not established.\n\nThird, and most serious, is the circularity. Theorem 1 assumes ||epsilon_k|| <= epsilon0 e^{-lambda t_k}, i.e., the NN approximation error decays exponentially. Nothing in Assumption 4, the training procedure, or universal approximation supplies this. Universal approximation is static over a compact set; it gives no pointwise-in-time decay along a trajectory. So the theorem's advertised exponential convergence is essentially assumed, not proved. Corollary 1 retreats to ultimate boundedness, which is the honest statement.\n\nFourth, the training loss (10)-(13) is inconsistent with the observer (3). MSEd = ||\\hat x_{k+1} - f(\\hat x_k) - B u_k||^2 omits the correction term L_k(y_k-\\hat y_k) and the SMC term nu_k. Minimizing it pushes the observer toward open-loop prediction and suppresses the very correction the NN is supposed to learn. That alone undermines the claim that the network learns a stabilizing gain.\n\nThe paper also cites related work fairly, so no issue there, but the absence of code/data makes it hard to verify the empirical claims.\n\nVerdict: I would not trust the theorem, and the training loss needs redesign. The idea might be salvageable with a corrected loss and a Lyapunov analysis that treats the NN error as a bounded disturbance (giving ultimate boundedness, not exponential convergence). As it stands, this is not a sound paper. If it comes to a journal, I'd send it back for major revision only on the condition that the authors fix the loss and either prove a valid sector condition or drop the exponential claim. For an arXiv-level reading group, it's a useful cautionary example of how learning-based observer papers can overclaim.\n\nRecommendation: do not send to peer review in current form; desk reject is defensible. If the authors revise, it could become a legitimate contribution, but the current version is not there.","headline":"A useful engineering idea that overclaims its theory: the convergence proof is circular and the training loss suppresses the observer correction.","tokens_in":17229,"tokens_out":4908,"would_cite":false,"duration_ms":46958,"reading_group":"yes","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93B53","93C10","93D05"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that a neural network can learn a time-varying observer gain for a general nonlinear discrete-time system, and that coupling this learned gain with an adaptive sliding-mode correction drives the state-estimation error…","keywords":["nonlinear state estimation","software sensor","neural network observer","adaptive sliding mode control","time-varying observer gain","physics-informed loss","Lyapunov convergence","three-tank system"],"falsifier":"Record $\\|L_k-L_k^*\\|$ during the reported simulations, or on a test system with a known optimal gain, and check whether the measured error falls on or below an exponential envelope $\\varepsilon_0 e^{-\\lambda t_k}$; if the error remains above a positive constant, Theorem 1’s central hypothesis is violated and the observed convergence would need another explanation. A complementary check is to evaluate the sufficient condition (33) step by step, noting that its right-hand side contains $\\|e_k\\|$, so the certificate cannot be verified before the error trajectory is known.","tokens_in":16203,"feed_emoji":"📡","tokens_out":11263,"duration_ms":111116,"temperature":0.7,"pith_summary":"This paper tries to establish a “nearly universal” software sensor: a state observer that works for nonlinear dynamical systems in their most general form, using only sensor outputs and known inputs, without a linearization step or an explicit transformation to a linear system. The proposed observer lets a neural network produce a time-varying gain matrix from the current time, input, and measured output, while an adaptive sliding-mode term corrects the output error. The authors prove that if the network’s approximation of an ideal stabilizing gain decays exponentially and the sliding-mode gain stays above a positive threshold, the estimation error converges exponentially to a bounded region around zero. The payoff would be an observer-design recipe that applies where classical observers struggle, including systems with non-differentiable dynamics and outputs that mix several states, and the training does not require ground-truth state trajectories.","feed_headline":"Neural-network observer tracks nonlinear states without linearization","feed_subtitle":"Trained only on sensor readings, the observer adapts in real time and works where linearization fails.","key_machinery":"The load-bearing mechanism is the observer update with three coupled pieces: a multilayer feedforward network with tanh activations outputs the gain matrix $L_k$ from $(t_k,u_k,y_k)$; the adaptive sliding-mode term $\\nu_k=-K_k\\tanh(y_k-\\hat{y}_k)$ with $K_k=K_0+\\alpha\\|y_k-\\hat{y}_k\\|^2$ gives a smooth, error-dependent correction whose gain grows with the output error; and a physics-based MSE loss (residual dynamics plus output error plus weight regularization) trains the network without ground-truth state trajectories. In the convergence proof, the key identity is the error recursion $e_{k+1}=(F_k-L_k^*H_k)e_k+w_k$, where the optimal gain $L_k^*$ is chosen so that $A_k^T P_{k+1}A_k-P_k\\le -\\gamma P_k$ with $A_k=F_k-L_k^*H_k$, and $w_k$ collects the neural-network error, higher-order Taylor terms, and the sliding-mode action; the inequality $e^T H^T\\tanh(He)\\ge \\sigma\\|He\\|^2$ absorbs the sliding-mode term into the Lyapunov decrease.","core_discovery":"The central claim is that the observer\n$$\n\\hat{x}_{k+1}=f(\\hat{x}_k)+L_k(y_k-\\hat{y}_k)+Bu_k+\\nu_k,\n\\qquad L_k=\\mathcal{N}(t_k,u_k,y_k),\n$$\nwith the adaptive sliding-mode term $\\nu_k=-K_k\\tanh(y_k-\\hat{y}_k)$ and gain $K_k=K_0+\\alpha\\|y_k-\\hat{y}_k\\|^2$, estimates the full state of a system whose dynamics and output map are only assumed bounded on a compact set and uniformly observable. The proof decomposes the learned gain as $L_k=L_k^*+\\varepsilon_k$, treats the approximation error $\\varepsilon_k$ as a disturbance, and uses a Lyapunov function to show that when $\\|\\varepsilon_k\\|\\le \\varepsilon_0 e^{-\\lambda t_k}$ and $K_k\\ge K_{\\min}>0$, the error $e_k=x_k-\\hat{x}_k$ converges exponentially to a bounded region; if the approximation error is only bounded, Corollary 1 gives ultimate boundedness with limiting size proportional to that bound. The same architecture is then demonstrated on chaotic, weakly observable, and non-differentiable benchmarks, including a three-tank system whose flow law contains signum and square-root terms, with the claim that such cases defeat linearization-based designs.","pith_inferences":["A reader could settle the practical reach of the theorem by recording $\\|L_k-L_k^*\\|$ during training on one of the paper’s benchmarks and checking whether it follows an exponential envelope; the paper supplies no architecture or training guarantee that it does.","The same physics-based loss could plausibly be adapted to output-feedback control, where the learned time-varying gain would close the loop as well as correct estimates.","Because the sliding-mode gain depends only on the output error, a natural extension is to asynchronous or fused sensor measurements; the loss already accepts any output map $h$.","The square-wave three-tank experiment suggests a stress test: introducing measurement dropout or sensor bias and observing whether the time-varying gain remains stable would probe the real-time adaptability claim beyond the reported simulations."],"forward_implications":["Observer synthesis no longer requires a state transformation or linearization: the gain matrix is produced directly from time, input, and output measurements.","Training runs without ground-truth state trajectories, using only sensor outputs and the system equations as a constraint, which matters when true states are unavailable.","With an exponentially decaying network approximation error, the paper guarantees exponential convergence of the estimation error to a bounded neighborhood; with only a bounded error, the error remains ultimately bounded by a constant times that bound.","The method is claimed to extend to non-differentiable dynamics such as the three-tank signum/square-root flow, and to measurements that are nonlinear functions or sums of states.","On the reverse Duffing benchmark, the paper reports lower MSE, RMSE, MAE, and SMAPE than supervised neural networks, unsupervised autoencoders, and supervised physics-informed neural networks, with and without noise."],"supporting_citations":[{"why":"Provides the reverse Duffing benchmark and the supervised physics-informed neural-network architecture whose network settings the paper reuses for its comparison.","marker":"[4]"},{"why":"Supplies the Neural ODE observer baseline whose 5-second stabilization time is compared with the paper’s 2.98 seconds in Example 3.","marker":"[5]"},{"why":"Source of the nonlinear academic system in Example 4 and the Riemannian-metric observer conditions the paper contrasts with its direct-gain approach.","marker":"[25]"},{"why":"Earlier adaptive sliding-mode observer for uncertain nonlinear systems that motivates the adaptive SMC correction term.","marker":"[11]"},{"why":"Unsupervised physics-informed neural-network observer using contraction analysis, cited as a state-of-the-art hybrid observer for comparison.","marker":"[17]"},{"why":"Deep neural-network integral sliding-mode control for fully unknown dynamics, cited as evidence for the NN-plus-SMC strategy the paper transfers to observation.","marker":"[22]"}],"fun_headline_variants":["Neural observer adapts to nonlinear systems without linearization","SMC-boosted neural observer tracks states with no linearization","Adaptive neural sliding-mode estimator handles nonlinearities","Observer learns from measurements only, resists noise and chaos"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the neural network’s error in reproducing the ideal time-varying observer gain shrinks exponentially as time increases; the paper assumes this decay (Assumption 4 plus the condition on $\\|\\varepsilon_k\\|$ in Theorem 1) rather than deriving it from the training procedure, so if the trained network’s error only stays bounded or grows, the exponential convergence guarantee does not follow.","fun_headline_variants_meta":{"raw":{"variants":["Neural observer adapts to nonlinear systems without linearization","SMC-boosted neural observer tracks states with no linearization","Adaptive neural sliding-mode estimator handles nonlinearities","Observer learns from measurements only, resists noise and chaos"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000208,"raw_usage":{"total_tokens":1471,"prompt_tokens":1081,"completion_tokens":390,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":697,"completion_tokens_details":{"reasoning_tokens":323}},"tokens_in":697,"tokens_out":390,"duration_ms":56461,"temperature":1.0,"reasoning_tokens":323,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T18:54:10.892272+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Record $\\|L_k-L_k^*\\|$ during the reported simulations, or on a test system with a known optimal gain, and check whether the measured error falls on or below an exponential envelope $\\varepsilon_0 e^{-\\lambda t_k}$; if the error remains above a positive constant, Theorem 1’s central hypothesis is violated and the observed convergence would need another explanation. A complementary check is to evaluate the sufficient condition (33) step by step, noting that its right-hand side contains $\\|e_k\\|$, so the certificate cannot be verified before the error trajectory is known.","supporting_citations":[{"cited_title":"Learning- based design of luenberger observers for autonomous nonlinear sys- tems,","cited_arxiv_id":null,"evidence_quote":"Provides the reverse Duffing benchmark and the supervised physics-informed neural-network architecture whose network settings the paper reuses for its comparison."},{"cited_title":"Learning robust state observers using neural odes,","cited_arxiv_id":null,"evidence_quote":"Supplies the Neural ODE observer baseline whose 5-second stabilization time is compared with the paper’s 2.98 seconds in Example 3."},{"cited_title":"Convergence of nonlinear observers on Rn with a riemannian metric (part i),","cited_arxiv_id":null,"evidence_quote":"Source of the nonlinear academic system in Example 4 and the Riemannian-metric observer conditions the paper contrasts with its direct-gain approach."},{"cited_title":"An adaptive sliding-mode observer for a class of uncertain nonlinear systems,","cited_arxiv_id":null,"evidence_quote":"Earlier adaptive sliding-mode observer for uncertain nonlinear systems that motivates the adaptive SMC correction term."},{"cited_title":"Unsupervised Physics-Informed Neural Network-based Nonlinear Observer design for autonomous systems using contraction analysis","cited_arxiv_id":"2411.09237","evidence_quote":"Unsupervised physics-informed neural-network observer using contraction analysis, cited as a state-of-the-art hybrid observer for comparison."},{"cited_title":"Design of a deep neural network-based integral sliding mode control for nonlinear systems under fully unknown dynamics,","cited_arxiv_id":null,"evidence_quote":"Deep neural-network integral sliding-mode control for fully unknown dynamics, cited as evidence for the NN-plus-SMC strategy the paper transfers to observation."}],"review_version":1}