{"id":"bc993b09-1ae3-4338-8b1b-80fe56123aec","arxiv_id":"2508.20367","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A neural operator approximate predictor with online delay adaptation yields semi-global practical stability for nonlinear systems with unknown constant input delay.","lead":"This paper proves that a learned neural network can replace the mathematical predictor in feedback controllers for nonlinear systems with an unknown, constant actuation delay, and still keep the system stable. The result turns an expensive online numerical computation into a fast network evaluation, with a 15x speedup shown on a biological activator/repressor model.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3's proof applies the neural operator error bound ε on a compact set K without proving the closed-loop trajectory stays in K; the uniform bound in (56) is therefore unsupported.","rationale":"The reader's weakest assumption identifies precisely the gap I find most load-bearing: the uniform approximation bound ε is stated for a fixed compact set K, but the proof applies it uniformly along the closed-loop trajectory without showing that the trajectory remains in K. This is not a manufactured concern; it is the step that connects the neural operator approximation theorem (Theorem 2) to the stability theorem (Theorem 3). If the trajectory leaves K, the bound in (56) fails, and the dissipation argument collapses. The paper also flags related missing support in its own text: it notes in Section III that the number of parameters/data required for the bound is beyond its scope, and it does not provide a reachability analysis. I find the numerical results promising and the high-level approach plausible, and I credit the use of a standard universal approximation theorem and a Lyapunov-Krasovskii framework. However, the proof as written contains a circularity: the approximation bound is known only on a set that must contain the trajectory, yet the trajectory bound is derived only after using the approximation bound. The reader's verdict of CONDITIONAL is appropriate: the central idea is credible, but the proof needs an explicit construction of K (or an alternative a priori bound on the reachable set) before the main theorem can be considered fully established. I do not see a reason to strengthen to REJECT, because the gap may be patchable with standard semi-global arguments; and I do not see a reason to downgrade to ACCEPT, because the gap is in the main theorem, not a peripheral lemma.","tokens_in":13274,"tokens_out":2979,"duration_ms":33804,"concrete_test":"Attempt to construct K explicitly from (32) and the stated assumptions: (i) set K = {(X, T_φ U, φ) : |X| ≤ R_X, ‖U‖_{L∞[0,1]} ≤ R_U, φ ∈ [D, \\bar D]} with R_X, R_U to be determined; (ii) using only the exact-predictor Lyapunov estimates (44)–(46) and the claimed final bounds, derive a priori bounds on |X(t)|, ‖u(t)‖_{L∞}, and \\hat D(t) that hold for all t without invoking (56); (iii) check whether these bounds are contained in K. If the a priori bounds depend on the same ε whose validity requires K, the argument is circular.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is Theorem 3, which asserts semi-global practical stability under the approximate predictor (28). The proof's key step is equation (56): (κ(P(x,t))−κ(ˆp(x,t)))² ≤ M₃² ε² for all x∈[0,1] and t≥0. This requires the uniform approximation guarantee of Theorem 2 to hold on the actual trajectory (X(t), T_{\\hat D(t)}U, \\hat D(t)) for all time. However, Theorem 2 only supplies a bound on a fixed compact set K, and the paper never constructs K from the initial-condition constraint (32), nor proves that the reachable set of the closed loop is contained in K. The semi-global claim would require an argument like: for any prescribed compact set of initial conditions, choose K large enough and ε small enough so that the trajectory never leaves K; but the reachable set depends on the controller, which depends on the trained neural operator, and the approximation bound on K is only known after fixing K. No forward-invariance or a priori trajectory bound is established without already assuming (56). The citation to [24, Theorem C.3] for (57) also assumes a uniformly bounded perturbation term, which is exactly what fails if the trajectory exits the training region. Assumption 4 gives Lipschitz growth but not boundedness, so escape from any fixed K is not excluded. Thus the proof is circular at the point where the trajectory enters and remains in K; without this, inequalities (50), (57), and the final bounds (33)–(35) do not follow. This is a missing-support issue in the proof, not merely a disagreement with an existing convention.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a delay-adaptive predictor feedback controller for nonlinear systems with an unknown constant actuator delay, in which the analytically intractable predictor is replaced by a trained neural operator approximation. The main theoretical claim, Theorem 3, asserts semi-global practical stability of the closed-loop system under a Lyapunov-Krasovskii analysis, with the ultimate bound depending on the neural operator approximation error ε and the delay uncertainty ΔD. The paper also reports numerical experiments on a biological activator/repressor system, showing a 15x speedup over a numerical predictor.","tokens_in":13617,"tokens_out":4309,"duration_ms":39255,"significance":"If the proof were complete, this would be the first stability result for approximate predictor feedback with unknown delay, extending the known-delay analysis of [6] and providing a rigorous framework for neural operator approximation in adaptive delay systems. The theorem is appropriately conditional on ε and ΔD, and the authors correctly emphasize that the adaptive law achieves practical stability rather than parameter convergence. The availability of code and the systematic experimental comparison are strengths. However, a load-bearing step in the proof of Theorem 3 is not supported: the uniform neural operator approximation bound is invoked on the closed-loop trajectory without establishing that the trajectory remains in the compact set on which the approximation is valid.","major_comments":[{"comment":"The inequality (κ(P(x,t))−κ(ˆp(x,t)))² ≤ M₃² ε² is asserted for all x∈[0,1] and t≥0, but Theorem 2 only guarantees such a uniform bound on a fixed compact set K⊂X×C²([0,1];U)×D. The paper never constructs K from the initial-condition constraint (32), nor does it prove that the closed-loop trajectory (X(t), T_{D̂(t)}U, D̂(t)) remains in K for all time. Since Assumption 4 gives only Lipschitz growth and not boundedness, escape from any fixed K is not excluded. This is not a minor technicality: (56) is used to obtain (50), then (57), and ultimately the stability bounds (33)–(35). Without a forward-invariance or a priori trajectory bound, the proof's central claim is unsupported.","section":"§IV, Theorem 3 proof, Eq. (56)"},{"comment":"The application of [24, Theorem C.3] requires the perturbation term to be uniformly bounded in time (or to satisfy the hypotheses of that theorem). Here the perturbation is (κ(P)−κ(ˆp))², whose boundedness rests entirely on the unproven uniform bound (56). Moreover, the sentence 'when ε and ΔD are small enough relative to all possible values of W' is vague: the condition ε+(ΔD)≤α₄(W(t))≤α₅(X+U+D) appears to presuppose a bound on W(t) that has not been established before the stability estimate. The logical structure is thus circular at this point; the smallness conditions need to be stated explicitly in terms of initial conditions and proven prior to invoking Theorem C.3.","section":"§IV, Eq. (57)"},{"comment":"There is a notational and conceptual confusion: Γ(t) is defined in (30) as a Lyapunov functional, but in (31) and in the proof the same symbol Γ is used for the scalar bound X+U+D. Then (32) imposes Γ(0)≤α₃*(Γ−α₁*(ε)−α₂*(ΔD)). The semi-global claim would require that for any prescribed compact set of initial conditions one can choose the compact set K (on which the neural operator is trained) and ε* so that the trajectory remains in K and (32) holds. Because the controller depends on the trained neural operator, the reachable set depends on the approximation itself, and the paper does not show how K is chosen before the controller is designed. This is a genuine gap in the semi-global statement; the authors should either provide an explicit construction of K from the initial-condition set or weaken the claim to a local or regional result.","section":"§IV, Theorem 3 statement and proof after Eq. (59)"}],"minor_comments":[{"comment":"In (56), the constant M₃ is used as a Lipschitz constant for κ, but Assumption 4 defines M₃ as a growth bound (|κ(X)|≤M₃|X|) and M₄ as the derivative bound (|dκ/dX|≤M₄). The Lipschitz bound in (56) should use M₄, not M₃.","section":"§II, Assumption 4 and Eq. (56)"},{"comment":"Several integrals in the proof of Lemma 1 have a lower limit of −1 where they should be 0 (for example, the integrals in the lines following Eq. (24) and Eq. (26)). This appears to be a typo but should be corrected for readability.","section":"Lemma 1 proof, Eq. (24)–(26)"},{"comment":"The symbol Γ is used both for the Lyapunov functional in (30) and for the scalar bound X+U+D in (31) and in the proof. These should be denoted differently to avoid confusion.","section":"Notation"},{"comment":"In Assumption 3, X and U are used both for the state/input variables and for the compact domains; similarly, Theorem 2's notation K⊂X×C²([0,1];U)×D mixes sets and variables. Using calligraphic letters or explicit domain names would clarify the statements.","section":"Assumption 3 and Theorem 2"},{"comment":"The expression in (55) contains a term 1+e^{b/(Dγ)}(ΔD)² whose placement within the parentheses is ambiguous; adding brackets would improve readability.","section":"Eq. (55)"}],"recommendation":"major_revision","confidential_remarks":"The central idea is plausible and the paper is likely to be of interest to the delay-systems and operator-learning communities, but the proof of Theorem 3 has a load-bearing gap concerning the domain of the neural operator approximation. The gap appears fixable within the manuscript's scope, for example by adding a boundedness/comparison argument that constructs the compact set from the initial-condition set, or by explicitly stating a local version of the theorem. I would encourage the editor to request a major revision rather than rejection, provided the authors can close the trajectory-in-K argument and make the use of the comparison theorem in (57) rigorous."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a real step forward in combining neural operator approximations with delay-adaptive predictor feedback, but the main theorem as written has a missing piece that the authors need to close before the result is established.\n\nWhat's new and good: The paper takes the known-delay neural operator predictor from [6] and the exact delay-adaptive predictor from [9] and puts them together. The continuity lemma for the predictor operator is clean and useful. The stability analysis follows the standard Lyapunov-Krasovskii backstepping template and correctly identifies that the approximation error enters additively, leading to a practical stability result rather than asymptotic. The biological example is a reasonable sanity check, and the 15x speedup at fine discretization is plausible.\n\nSoft spots: The gap is in Theorem 3's proof. Equation (56) asserts the approximation error bound holds for the actual closed-loop trajectory for all time. But Theorem 2 only gives a uniform bound on a fixed compact set K. The paper never constructs K from the initial-condition constraint (32), nor proves the trajectory stays in K. The same issue appears with Assumption 3's local Lipschitz bound: the closed-loop state may leave the compact domains on which Cf is defined. So the proof assumes what it needs to show: that the reachable set is contained in the region where the error and Lipschitz bounds are valid. This is a genuine missing-support problem, not a stylistic quibble. The jump to [24, Theorem C.3] also needs checking once the perturbation term is not known to be uniformly bounded.\n\nSecondary issues: the proof leans heavily on [9] for bounds (44)-(46) without reproducing them, and the experiments report no error bars or measured approximation error, only two training checkpoints.\n\nThat said, the gap looks patchable. A semi-global argument could choose K large enough and then show via the stability bound that trajectories from a prescribed initial set stay in K, using the error bound on K. The authors should be asked to supply such an argument, or to state the theorem as a local result.\n\nBottom line: worth a serious referee. I would send it out, with a request for a major revision that fixes the invariance/forward-completeness issue. Readers interested in delay systems or operator learning in control will want to read this, even if the theorem needs tightening.","headline":"A worthwhile combination of neural operator predictors and delay adaptation, but the main theorem's proof assumes the trajectory stays in the training region without proving it.","tokens_in":14131,"tokens_out":4037,"would_cite":true,"duration_ms":37510,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93C23","93C10","68T07"],"pacs":[],"model":"deepseek-v4-flash","headline":"Neural operator predictors can stabilize nonlinear systems whose actuator delay is unknown, with practical convergence controlled by approximation error and delay uncertainty.","keywords":["neural operators","predictor feedback","actuator delay","delay-adaptive control","nonlinear systems","Lyapunov-Krasovskii functional","semi-global practical stability","operator learning"],"falsifier":"Train a neural operator predictor on a compact set $K$, run the closed loop (28)-(12) from an initial condition satisfying (32), and compute along the trajectory the actual maximum error $\\sup_{x,t}|P(X, T_{\\hat D}U, \\hat D)-\\hat P(X, T_{\\hat D}U, \\hat D)|$. If this error exceeds the $\\epsilon$ used in Theorem 3 while the state leaves the predicted residual neighborhood, the missing construction of $K$ from (32) is the cause; if stability persists anyway, the theorem's compact-set condition is conservative.","tokens_in":13086,"feed_emoji":"🧠","tokens_out":7930,"duration_ms":70841,"temperature":0.7,"pith_summary":"This paper claims that predictor feedback for nonlinear systems with unknown, arbitrarily long actuator delays can be safely implemented with a learned neural operator instead of an exact predictor. It proves that when the neural operator approximates the predictor uniformly to within $\\epsilon$ on a compact set, and the delay estimate is updated online with a projection law, the closed-loop state converges to a neighborhood whose radius is controlled by $\\epsilon$ and by the size of the delay-uncertainty interval. This matters because exact nonlinear predictors are implicit ODEs with no closed form, while numerical approximations are slow; a fast learned predictor with a stability certificate makes predictor feedback practical for real-time control. The result is a semiglobal practical stability theorem, not a global asymptotic one: the residual set is nonzero for any finite training error or nonzero delay uncertainty.","feed_headline":"Neural operators stabilize unknown-delay nonlinear systems","feed_subtitle":"A new proof ties the convergence radius to predictor error and delay-estimate uncertainty.","key_machinery":"The argument is carried by the predictor operator $P(X,U,\\varphi)(s)=X+\\varphi\\int_0^s f(P(s),U(s))ds$, the transport-PDE representation of the delay ($D u_t=u_x$, $u(1,t)=U(t)$), the backstepping transformation $w(x,t)=u(x,t)-\\kappa(P(x,t))$, and the Lyapunov-Krasovskii functional $W(t)=D\\log N(t)+\\frac{b}{\\gamma}\\tilde D(t)^2$ with $N(t)=1+V(X)+b\\int_0^1(1+x)w(x,t)^2dx$. The universal approximation theorem for neural operators enters by guaranteeing a predictor $\\hat P$ with uniform error $\\epsilon$ on a compact set; this turns the approximate predictor into an additive perturbation that the Lyapunov analysis bounds in terms of $\\epsilon$ and the projection bounds $\\underline D, \\overline D$.","core_discovery":"Under the plant $\\dot X = f(X, U(t-D))$ with $D$ unknown in $[\\underline D, \\overline D]$, and under Assumptions 1–4, the paper proves that the controller $U(t)=\\kappa(\\hat P(X(t), T_{\\hat D(t)}U, \\hat D(t)))$ with the projection-based adaptive law (11) makes the closed loop semiglobally practically stable. Theorem 3 gives constants $\\gamma^*, b^*, \\epsilon^*$ and class-$\\mathcal K_\\infty$ and $\\mathcal{KL}$ bounds such that, for $\\gamma<\\gamma^*$, $b>b^*$, $\\epsilon<\\epsilon^*$, and initial states satisfying (32), $\\Gamma(t)\\le\\alpha_3^*(\\Gamma(0))+\\alpha_1^*(\\epsilon)+\\alpha_2^*(\\Delta D)$ and $|X(t)|^2\\le\\beta_1^*(|X(0)|^2,t)+\\alpha_4^*(\\Delta D)+\\alpha_5^*(\\epsilon)$. The residual set shrinks as the neural operator's uniform approximation error $\\epsilon$ and the delay-uncertainty window $\\Delta D = \\overline D - \\underline D$ go to zero, giving a stability guarantee for a learned predictor rather than an exact one.","pith_inferences":["The uniform-error assumption is the fragile point: Theorem 2 fixes a compact set $K$ before the closed loop is run, and the paper never derives $K$ from the initial-condition constraint (32), so a trajectory that leaves the training region would invalidate the bound $|P-\\hat P|\\le\\epsilon$ used at (56).","If the computational gain grows with the cost of evaluating $f$, then applying this scheme to more expensive dynamics than the simple Hill-function example should reproduce or exceed the larger speedups reported for known-delay neural predictors.","A direct extension would be to state-dependent or slowly time-varying delays; the current projection law is constant-delay-specific, and parameter convergence is not established, so the adaptive estimate may not track a changing delay.","Because the controller uses the full distributed actuator measurement $u(x,t)$, extending the result to output-feedback settings with only boundary measurements would require an observer or a different measurement assumption."],"forward_implications":["A neural operator trained once offline can replace numerical predictor integration online without forfeiting a stability guarantee, for any plant satisfying the four assumptions.","Because the predictor error enters additively, the user can tune the practical convergence radius by choosing the training tolerance $\\epsilon$ and the delay bounds $\\underline D,\\overline D$.","The stability proof treats the neural operator as a black box meeting a uniform error bound, so the same guarantee covers any sufficiently accurate learned predictor, not only the architectures tested.","In validation on an activator/repressor clock, the learned predictor stabilizes the unstable equilibrium and runs about 15 times faster than numerical predictor integration at the finest discretization."],"supporting_citations":[{"why":"Supplies the delay-adaptive law, the backstepping transformation, and the Lyapunov framework that this paper extends to approximate predictors.","marker":"[9]"},{"why":"Provides the universal approximation theorem for neural operators (Theorem 1) that guarantees a predictor approximation with uniform error $\\epsilon$.","marker":"[27]"},{"why":"Established neural operator approximate predictors for nonlinear delay systems with known delay, the setting this paper extends to unknown delays.","marker":"[6]"},{"why":"Documents numerical predictor implementations and their stability guarantees, the computational baseline the neural operator replaces.","marker":"[18]"},{"why":"Introduced the transport PDE representation of the actuator delay used to reformulate the plant.","marker":"[22]"},{"why":"Supplies the comparison-principle argument used to convert the Lyapunov derivative bound into $\\mathcal{KL}$ and class-$\\mathcal K_\\infty$ stability estimates.","marker":"[24]"},{"why":"Defines the DeepONet architecture and operator universal approximation theorem used in training and validation.","marker":"[30]"},{"why":"Defines the Fourier Neural Operator architecture also used in the numerical validation.","marker":"[29]"},{"why":"Provides the biological activator/repressor model used for the numerical validation.","marker":"[11]"}],"fun_headline_variants":["Neural predictors stabilize systems with unknown delay","Offline-trained neural nets handle arbitrarily long actuator delays","Stability set shrinks with predictor error and delay uncertainty","Learned predictor control proves semiglobal practical stability"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes the trained neural operator's approximation error is bounded by $\\epsilon$ uniformly on a compact set that contains every state, input history, and delay estimate the closed loop will actually visit, but the paper never derives that set from the initial-condition constraint, so trajectories leaving the training region would break the key bound (56).","fun_headline_variants_meta":{"raw":{"variants":["Neural predictors stabilize systems with unknown delay","Offline-trained neural nets handle arbitrarily long actuator delays","Stability set shrinks with predictor error and delay uncertainty","Learned predictor control proves semiglobal practical stability"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001327,"raw_usage":{"total_tokens":5398,"prompt_tokens":940,"completion_tokens":4458,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":556,"completion_tokens_details":{"reasoning_tokens":4396}},"tokens_in":556,"tokens_out":4458,"duration_ms":26519,"temperature":1.0,"reasoning_tokens":4396,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T16:46:33.907250+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Train a neural operator predictor on a compact set $K$, run the closed loop (28)-(12) from an initial condition satisfying (32), and compute along the trajectory the actual maximum error $\\sup_{x,t}|P(X, T_{\\hat D}U, \\hat D)-\\hat P(X, T_{\\hat D}U, \\hat D)|$. If this error exceeds the $\\epsilon$ used in Theorem 3 while the state leaves the predicted residual neighborhood, the missing construction of $K$ from (32) is the cause; if stability persists anyway, the theorem's compact-set condition is conservative.","supporting_citations":[{"cited_title":"Delay-Adaptive Control for Nonlinear Systems,","cited_arxiv_id":null,"evidence_quote":"Supplies the delay-adaptive law, the backstepping transformation, and the Lyapunov framework that this paper extends to approximate predictors."},{"cited_title":"Karafyllis and M","cited_arxiv_id":null,"evidence_quote":"Documents numerical predictor implementations and their stability guarantees, the computational baseline the neural operator replaces."},{"cited_title":"Input Delay Compensation for Forward Complete and Strict-Feedforward Nonlinear Systems,","cited_arxiv_id":null,"evidence_quote":"Introduced the transport PDE representation of the actuator delay used to reformulate the plant."},{"cited_title":"Krstic, P","cited_arxiv_id":null,"evidence_quote":"Supplies the comparison-principle argument used to convert the Lyapunov derivative bound into $\\mathcal{KL}$ and class-$\\mathcal K_\\infty$ stability estimates."},{"cited_title":"Fourier Neural Operator for Parametric Partial Differential Equations,","cited_arxiv_id":null,"evidence_quote":"Defines the Fourier Neural Operator architecture also used in the numerical validation."},{"cited_title":"Design and analysis of an activator-repressor clock in e. coli,","cited_arxiv_id":null,"evidence_quote":"Provides the biological activator/repressor model used for the numerical validation."}],"review_version":2}