{"id":"62f21afa-7175-4e23-8c06-f119503bc7d3","arxiv_id":"2505.18008","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"MS-DeepONet computes one-shot multi-step predictions for nonlinear MIMO systems in model predictive control, with a proven universal approximation property and better benchmark performance than the standard DeepONet.","lead":"The paper introduces MS-DeepONet, a neural network architecture that predicts multiple future outputs of a nonlinear dynamic system in a single forward pass, which is well suited for model predictive control. The authors prove a universal approximation result for this architecture and show in simulations that it outperforms a standard DeepONet predictor on three nonlinear benchmark systems.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Universal-approximation proof lacks a bridge from Theorem 2 (scalar, output-specific trunk) to MS-DeepONet's shared trunk serving all N*n_y outputs.","rationale":"The reader's weakest_assumption is the full-state-feedback requirement (Remark 5). That is a real scope limitation, but it does not bear on the paper's central theoretical claim of universal approximation. The reader's rationale also lists 'the bridge from the universal approximation theorem to the exact architecture is a remark rather than a full proof' as one of several issues, but it was not elevated to the weakest assumption. I partial-agree with the reader: the proof gap is the load-bearing concern because the claimed theorem is the paper's main novelty. My proposed test is analytical: write out the multi-output extension of Theorem 2 for a shared-trunk architecture. If the extension is valid, the paper's central claim is correct and the concern is merely a presentation gap; if invalid, the paper's main contribution is unsupported. The verdict remains CONDITIONAL because the proof gap is fillable with additional exposition and because the empirical claims, while overstated for the pendulum case, do not invalidate the architecture itself. I therefore recommend UNCHANGED relative to the reader's conditional verdict.","tokens_in":23552,"tokens_out":20018,"duration_ms":199980,"concrete_test":"Re-derive the MIMO universal approximation statement for the exact MS-DeepONet architecture. Concretely: let G: K1 -> C(K2)^m with m=N*n_y. Show that for any epsilon>0 there exist p, a shared trunk t_i(x) (i=1..p), and branch outputs b_{r,i}(u) (r=1..m, i=1..p) such that max_{u,x} |G_r(u)(x) - sum_i b_{r,i}(u)t_i(x)| < epsilon. Verify the construction explicitly, e.g., by taking p=m*p', grouping the trunk functions into m blocks, and setting b_{r,i}=0 for i not in block r. Check that these zero-padded branch coefficients are continuous functions on K1 and are representable by the branch network (12). If the construction goes through, the gap is expositional and the claim holds; if the zero-padding or continuity fails, the central theorem is false.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim, stated in the abstract and Remark 1, is that MS-DeepONet is a universal approximator for multi-step prediction. The proof route is Theorem 2 (Section III), which guarantees, for a single scalar operator G(u)(x), an approximation sum_{k=1}^p [sum_{i=1}^n c^k_i sigma(xi^k_i*u+theta^k_i)] sigma(w^k*x+zeta^k). In that construction, the trunk functions sigma(w^k*x+zeta^k) are chosen specifically for the one scalar output being approximated. The MS-DeepONet architecture of Section III-A instead uses a single trunk network with p outputs t_i(x_k) shared by every one of the N*n_y output components, while the branch network has p*n_y*N outputs b^q_{i,j}(u_k). Remark 1 asserts that 'from Proposition 1 and Theorem 2 it follows' that such branch and trunk networks exist, but the proof never shows how a shared trunk can simultaneously serve all output components. A standard fix is to enlarge p to N*n_y*p' and zero-pad the branch outputs, but this construction is absent. If that construction cannot be made to work, the universal approximation property for the exact architecture, and hence the main theoretical contribution, is unproven. This is more load-bearing than the full-state-feedback restriction, which limits applicability but does not touch the representational claim, and more central than the missing pendulum baseline, which weakens only the empirical outperformance claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes the MS-DeepONet, an adaptation of the DeepONet architecture that maps an initial state x(kT_s) and a length-N piecewise-constant input sequence to all N future outputs of a continuous-time MIMO system in a single forward pass. One branch network encodes the stacked input sequence and one trunk network encodes only the state; the branch output dimension is p·n_y·N, so the product layer produces the complete multi-step output vector. The authors state a universal-approximation result for the architecture (Section III, Theorem 2 and Remark 1), derive an equivalent Kronecker-product basis representation (Lemma 1), propose a grid-search hyperparameter ablation (Algorithm 2), formulate MPC problems for both the standard and MS-DeepONet predictors (Problems 3 and 4), and validate the approach on the van der Pol oscillator, the quadruple tank process, and a cart-pendulum swing-up task. The implementation is released on GitHub.","tokens_in":23857,"tokens_out":15238,"duration_ms":157038,"significance":"If the universality claim is fully established, the architecture is practically attractive: it avoids repeated network evaluations and separate branch networks per input, and the basis representation points toward data-enabled predictive control. Theorem 2 is a credible adaptation of the Chen-and-Chen result, and the numerical protocol is fair: both architectures are trained on identical data with a grid search, and the reported losses, AME, and computation times support the comparison on the first two benchmarks. The publicly available PyTorch code and the explicit ablation study are strengths. The main weaknesses are that the step from Theorem 2 to the exact MS-DeepONet architecture is asserted in Remark 1 rather than proved, that the basis representation in Lemma 1 contains a Kronecker-ordering inconsistency, and that the claimed consistent outperformance on the pendulum benchmark is not supported by a standard-DeepONet baseline. These issues are repairable but currently prevent the paper from supporting its strongest claims.","major_comments":[{"comment":"Remark 1 is the only place where universality of the exact MS-DeepONet architecture is claimed, but it is asserted rather than proved. Theorem 2 supplies, for each scalar output component G_{q+(j-1)n_y}, an approximation whose trunk basis functions σ(w^k·x+ζ^k) are chosen for that component. In Eq. (14), however, all output components must share the same p trunk functions t_i(x). To close the gap one must construct a single p-dimensional trunk and p·n_y·N branch outputs, for example by enlarging p to the total number of basis terms and zero-padding the branch matrix B(u_k), and then verify that feedforward networks of the form (12)-(13) can realize the resulting representations. This construction is absent; as written, the proof does not establish the abstract's claim that the MS-DeepONet is a universal approximator for multi-step sequence prediction.","section":"Section III-A, Remark 1 (Eq. (15))"},{"comment":"The proof of Lemma 1 expands the first term of (21) as vec^T(Wb_i^T Wt)(Φ_t(x_k)⊗Φ_b(u_k)), but the basis vector Φ⊗ in Eq. (22) uses Φ_b(u_k)⊗Φ_t(x_k) in its first block. With the column-major vec defined in the Notation, vec^T(A)(v⊗u) equals u^T A v while vec^T(A)(u⊗v) equals v^T A^T u, so the two orderings are not interchangeable. Since Eq. (23) lists vec^T(Wb_i^T Wt) as the coefficient of Φ_b⊗Φ_t, the expression does not equal (21) as written; a Kronecker permutation must be introduced. This inconsistency affects the basis representation used in Remark 3 and the data-enabled formulation sketched in Remark 6.","section":"Section III-B, Lemma 1 (Eqs. (22)-(23))"},{"comment":"The abstract and conclusions state that MS-DeepONet consistently outperforms the standard DeepONet across the three benchmarks, but the pendulum-on-cart experiment contains no standard DeepONet baseline. Table 3 reports only an MS-DeepONet column for the pendulum, and Section VI-C evaluates no comparison against Problem 4. The consistently-outperforms claim is therefore supported only for the van der Pol and quadruple-tank examples; either add the standard DeepONet comparison or qualify the claim to those two examples.","section":"Section VI-C and Table 3"},{"comment":"Theorem 2 assumes G is a continuous operator from a compact subset of R^l into C(K2), but Section III does not state the assumptions on f and h under which the multi-step operator in Eq. (6) is continuous. Standard well-posedness of the system (1) would supply this, but the hypotheses (for example, Lipschitz continuity of f and continuity of h) and a short argument are missing. Without them, Theorem 2 cannot be invoked for the operator (6).","section":"Section III, Eq. (6) and Theorem 2"}],"minor_comments":[{"comment":"The caption says the output is the predicted multi-step output sequence \\bar{u}_0; this should be \\bar{y}_0 or the equivalent output notation, since the architecture predicts outputs, not inputs.","section":"Figure 4 caption"},{"comment":"After defining t(x_k, Ξ_i, Θ_i^t), the text repeats the branch definitions: 'Note in this case that Φ_b(u_k)=h^b_{l_i^b} and h^b_0=u_k' should instead state the trunk analogue, i.e., Φ_t(x_k)=h^t_{l_i^t} and h^t_0=x_k.","section":"Section IV-B"},{"comment":"In the proof of Proposition 1, the trunk argument is written as σ(w^k·u+ζ^k), but it should be σ(w^k·z+ζ^k) to match the operator input z; the surrounding text also contains a few typos such as 'from(6)' and 'MS-DeepoNet'.","section":"Appendix, Proposition 1 proof"},{"comment":"The statement that the MS-DeepONet MPC controller achieved performance comparable to the model-based MPC controller, with results omitted due to space limitations, is not verifiable; either include the comparison in the paper or remove the claim.","section":"Section VI-B"},{"comment":"The acronym AME is used in Table 3 but never defined; please define it (for example, absolute mean tracking error) and state over which trajectories and time horizons it is computed, especially for the pendulum swing-up task.","section":"Section VI and Table 3"},{"comment":"The full-state-feedback assumption is disclosed in Remark 5, but all benchmark controllers rely on it and the alternative input-output history variant is not implemented; a sentence in the introduction or abstract stating that the method currently requires full-state feedback would set expectations more accurately.","section":"Section V-A, Remark 5"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is likely of interest to the neural-MPC community, and the reproducible code, fair ablation, and open comparison on the first two benchmarks are genuine strengths. The main issues are the unproved bridge in Remark 1, the Kronecker-ordering inconsistency in Lemma 1, and the missing pendulum baseline; all three are repairable without changing the scope of the work, which is why I recommend major revision rather than rejection. I have no concerns about the novelty of the architecture itself, but the universal-approximation claim and the basis representation need to be correct before the paper can be accepted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The MS-DeepONet architecture is a real contribution and the empirical comparison is careful; but the universal approximation proof as written has a gap that needs closing before the main theoretical claim is solid.\n\nWhat's new: the paper replaces the multiple branch networks of standard DeepONet/MimoONet with one branch network that encodes the whole input sequence, and one trunk network that takes the current state; the output layer gives all N time steps for all n_y outputs in one forward pass. That's a sensible design for MPC. The basis representation in Lemma 1 and Remark 3 is also useful—it shows the predictor is linear in a feature map, which is exactly what you want for embedding in an optimization problem. The ablation study is a fair comparison: same data, same MPC costs, grid search over hyperparameters. On the van der Pol and quadruple tank examples, MS-DeepONet gets lower training and validation loss, lower tracking error, and trains 3-4x faster. That's credible.\n\nThe soft spot is the proof. Theorem 2 adapts Chen and Chen to operators from R^l to C(K2), and gives a sum over p trunk functions sigma(w^k*x+zeta^k) for one scalar output. MS-DeepONet uses the same p trunk outputs t_i(x) for every one of the N*n_y output components. Remark 1 says 'from Proposition 1 and Theorem 2 it follows' that the branch and trunk networks exist, but no construction is given. A reader can't tell how a single shared trunk can simultaneously serve all output components with arbitrary accuracy. The standard fix—enlarge p to N*n_y*p' and zero-pad the branch outputs—is not there. I suspect the claim is true and the fix is straightforward, but as written the universal approximation theorem is not proven for the exact architecture. That's a real gap, not a stylistic quibble.\n\nOther limitations are less serious. The full-state feedback assumption in Remark 5 is stated honestly, but it means the numerical results all assume x(kT_s) is measured; the input-output history alternative is not implemented. The pendulum experiment has no standard DeepONet comparison, so 'consistently outperforms' isn't supported there. And the model-based baseline for the quadruple tank is mentioned but omitted. These are addressable.\n\nWho this is for: people building learned predictors for MPC, and anyone interested in operator-learning architectures for control. It deserves peer review. I'd ask the authors to fix the proof gap (or clearly state the extra condition on p) and add or explicitly defer the missing comparisons. The architecture and empirical study are worth engaging with despite the incomplete proof.","headline":"A genuinely useful architecture and a fair empirical study, but the universal approximation proof has a missing bridge from scalar outputs to the shared-trunk multi-output setting.","tokens_in":24405,"tokens_out":3736,"would_cite":true,"duration_ms":36130,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["41A30","68T07","93C10","93C35"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that a modified DeepONet architecture computes all multi-step predictions of a nonlinear MIMO system in one forward pass and is a universal approximator for such mappings.","keywords":["model predictive control","DeepONet","operator learning","multi-step prediction","universal approximation","nonlinear MIMO systems","neural predictive control","basis representation"],"falsifier":"Train the MS-DeepONet and a standard DeepONet on a linear MIMO system with exactly computable sampled outputs, starting from the same data and matching widths, and compare multi-step prediction error to machine precision: for the universal-approximation claim to survive, the MS-DeepONet error must shrink toward zero as width and training epochs grow; a persistent error floor above the standard linear predictor's residual would falsify it.","tokens_in":23339,"feed_emoji":"🎛️","tokens_out":4924,"duration_ms":42675,"temperature":0.7,"pith_summary":"This paper tries to establish that a modified DeepONet architecture, the multi-step DeepONet (MS-DeepONet), can serve as the prediction model inside model predictive control for continuous-time nonlinear multi-input, multi-output systems. The authors prove that the MS-DeepONet is a universal approximator for the operator that maps an initial state and a piecewise-constant multi-step input sequence to the corresponding multi-step output sequence, so no recurrent network or iterative one-step simulation is needed to forecast a horizon. They also show empirically that, on the van der Pol oscillator, the quadruple tank process, and a cart-pendulum swing-up task, the MS-DeepONet learns faster, achieves lower validation loss, and yields lower mean tracking error than the standard DeepONet when both are embedded in MPC. If these claims hold, neural predictive control gains a one-shot, feedforward predictor with a formal representational guarantee.","feed_headline":"Multi-step DeepONet is a universal predictor for MPC","feed_subtitle":"A single forward pass forecasts an entire horizon, with a universal-approximation proof and gains on three benchmarks.","key_machinery":"The load-bearing object is the MS-DeepONet product-layer predictor $\\bar{y}_k = B^\\top(u_k) t(x_k)$, in which one branch network encodes the full multi-step input sequence $u_k = \\mathrm{col}(u_k,\\ldots,u_{k+N-1})$ and one trunk network encodes the measured state $x(kT_s)$; the branch output layer carries $p$ linear features per output per step, so a single forward pass produces all $N n_y$ future outputs. The proof machinery is a modified universal approximation theorem (Theorem 2) that replaces the infinite-dimensional function space of the classical operator theorem with a compact Euclidean input domain, so the operator $G$ maps compact input sequences directly to $C(K_2)$, and then shows the resulting one-layer network is a special case of the unstacked DeepONet. The supporting structural result is the basis representation $y_k = \\Theta_o \\Phi_\\otimes(u_k, x_k)$, obtained by expanding the product of affine branch and trunk layers into a Kronecker-feature vector, which turns the trained network into a linear-in-features predictor and connects it to data-enabled predictive control.","core_discovery":"The central claim is that the solution operator of a continuous-time MIMO system (1) restricted to piecewise-constant inputs and sampled outputs, written as $y_k = G(u_k)(x_k)$, can be approximated arbitrarily well by a single product-layer network: $\\sum_{i=1}^p b_{i,j}^q(u_k) t_i(x_k)$, where the branch net sees the entire multi-step input sequence $u_k$ and the trunk net sees only the initial state $x(kT_s)$. Theorem 2, adapting the Chen-Chen operator approximation theorem to the finite-dimensional domain of input sequences, is the proof that such a network exists for any $\\epsilon > 0$, and Remark 1 carries that result over to the MIMO operator (6). The paper further derives an equivalent basis form $y_k = \\Theta_o \\Phi_\\otimes(u_k, x_k)$ using a Kronecker product of branch and trunk features, which exposes the network as an adaptive linear basis and motivates a data-enabled predictive control variant. On the numerical side, the paper reports consistent improvements over the standard DeepONet/MimoONet predictor in training loss, validation loss, tracking error, and training time across three benchmark systems, including closed-loop swing-up data for the unstable pendulum on a cart.","pith_inferences":["If the full-state assumption (Remark 5) is relaxed to measured input-output histories, the same product-layer form should be trainable as an output-feedback predictor; the paper sketches this but gives no experiment, so a natural test is to re-run the three benchmarks with the history window replacing $x(kT_s)$.","The theorem's compact-domain premise suggests that the approximation guarantee degrades gracefully rather than catastrophically outside the training region; a testable extension would be to quantify worst-case multi-step error as the prediction horizon $N$ grows beyond the trained value and to compare with one-step-recursive DeepONet at the same total width.","The candidate basis might transfer between similar systems: features $\\Phi_t$ learned on one plant could be frozen and only $\\Theta_o$ re-fitted for a neighboring plant, which would turn MS-DeepONet into a reusable linear-in-features plant model for fast MPC re-tuning."],"forward_implications":["An MPC controller using MS-DeepONet solves Problem 3 with a single matrix-vector evaluation $B^\\top(u_k)t(x_k)$ per candidate input, avoiding recurrent unrolling and multiple branch networks; measured computation times stay below the sampling period in the presented benchmarks.","The universal approximation result means representational completeness: for any continuous-time nonlinear MIMO system with piecewise-constant inputs over a compact operating region, a sufficiently large MS-DeepONet can in principle represent the exact sampled multi-step input-output map, so prediction error within MPC is a training issue rather than an architectural ceiling.","The basis form $y_k = \\Theta_o \\Phi_\\otimes(u_k, x_k)$ yields a DeePC-style formulation by constructing a data matrix $\\Phi_\\otimes$ from learned features, with the predictor solving a regularized problem in the coefficient vector $g_k$; the paper proposes but does not implement this variant.","The architecture's lower training loss and up to several-fold shorter training time compared with standard DeepONet (55 vs 14 minutes on van der Pol; 164 vs 40 minutes on quadruple tank) suggest the one-shot multi-step parametrization is an easier learning problem."],"supporting_citations":[{"why":"Supplies the Tauber-Wiener operator approximation theorem (Theorem 3) and Theorem 5 that the authors adapt into their Theorem 2 for finite-dimensional input sequences.","marker":"[23]"},{"why":"Defines the DeepONet branch-trunk architecture whose multi-step MIMO extension is the paper's central object.","marker":"[1]"},{"why":"Provides the MimoONet extension used as the standard DeepONet baseline in the MIMO comparison and in the MPC formulation.","marker":"[32]"},{"why":"Introduces the basis-function viewpoint of feedforward predictors that the paper adapts to derive the candidate basis and the DeePC-style formulation.","marker":"[34]"},{"why":"Supplies the quadruple tank benchmark system used for the MIMO comparison.","marker":"[42]"},{"why":"Supplies the energy-based controller whose closed-loop swing-up trajectories form the training data for the pendulum on a cart.","marker":"[43]"}],"fun_headline_variants":["MS-DeepONet computes multi-step predictions in one shot","Universal approximation for multi-step DeepONet in MPC","Single-pass multi-step forecasting for model predictive control","MS-DeepONet outperforms standard DeepONet on three benchmarks","Multi-step DeepONet: one forward pass for full control horizon"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The method assumes the full state $x(kT_s)$ is measured at every sampling instant; if states are unobserved, the MS-DeepONet predictor as implemented cannot be applied and the numerical results do not cover that setting.","fun_headline_variants_meta":{"raw":{"variants":["MS-DeepONet computes multi-step predictions in one shot","Universal approximation for multi-step DeepONet in MPC","Single-pass multi-step forecasting for model predictive control","MS-DeepONet outperforms standard DeepONet on three benchmarks","Multi-step DeepONet: one forward pass for full control horizon"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000237,"raw_usage":{"total_tokens":1580,"prompt_tokens":1091,"completion_tokens":489,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":707,"completion_tokens_details":{"reasoning_tokens":405}},"tokens_in":707,"tokens_out":489,"duration_ms":3892,"temperature":1.0,"reasoning_tokens":405,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T14:37:08.145997+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Train the MS-DeepONet and a standard DeepONet on a linear MIMO system with exactly computable sampled outputs, starting from the same data and matching widths, and compare multi-step prediction error to machine precision: for the universal-approximation claim to survive, the MS-DeepONet error must shrink toward zero as width and training epochs grow; a persistent error floor above the standard linear predictor's residual would falsify it.","supporting_citations":[{"cited_title":"Universal approximation to nonlinear operators by neural networks with arbitrary activation functions and its applica- tion to dynamical systems,","cited_arxiv_id":null,"evidence_quote":"Supplies the Tauber-Wiener operator approximation theorem (Theorem 3) and Theorem 5 that the authors adapt into their Theorem 2 for finite-dimensional input sequences."},{"cited_title":"Learning nonlinear operators via DeepONet based on the universal approxima- tion theorem of operators,","cited_arxiv_id":null,"evidence_quote":"Defines the DeepONet branch-trunk architecture whose multi-step MIMO extension is the paper's central object."},{"cited_title":"Deep multi-input and multi-output operator networks method for optimal control of pdes,","cited_arxiv_id":null,"evidence_quote":"Provides the MimoONet extension used as the standard DeepONet baseline in the MIMO comparison and in the MPC formulation."},{"cited_title":"Neural data–enabled predictive control,","cited_arxiv_id":null,"evidence_quote":"Introduces the basis-function viewpoint of feedforward predictors that the paper adapts to derive the candidate basis and the DeePC-style formulation."},{"cited_title":"Four MPC implementations compared on the Quadruple Tank Process Benchmark: pros and cons of neural MPC,","cited_arxiv_id":null,"evidence_quote":"Supplies the quadruple tank benchmark system used for the MIMO comparison."},{"cited_title":"Swing-up and stabiliza- tion of a cart–pendulum system under restricted cart track length,","cited_arxiv_id":null,"evidence_quote":"Supplies the energy-based controller whose closed-loop swing-up trajectories form the training data for the pendulum on a cart."}],"review_version":1}