FAME: Adaptive Functional Attention with Expert Routing for Function-on-Function Regression

We thank the reviewer for the helpful suggestions. We have revised the paper to address all comments and respond point‑by‑point. Due to the rebuttal’s format and character limits, we present only newly added or key sections; full formatting and content will be finalized in the manuscript.

• Question (1): Your concern is fully justified, and we will include quantitative measurements of compute and memory usage. Cases 1–2 jointly span the two principal axes of computational load—the number of functions and the number of sampling points per function—so they are well suited for reporting cost, and all results will be produced on the same ordinary desktop (AMD Ryzen 9 5950X CPU) . In brief, basis/FPCA/kernel methods can train faster when grids are fixed and models are small, but they depend on pre‑specified bases or kernels and may fit particular data poorly; by contrast, FAME is fully data‑driven—while training is more costly, it remains lightweight in absolute terms , and inference time is comparable to classical baselines. Even in our most complex setting, training remains tractable (≈ 207 s/epoch × 20 epochs ≈ 69 min on this CPU), and post‑training inference time is on the order of seconds. Memory usage is even less of a concern and varies little across tasks, so we report average usage: the datasets are purely numeric and small on disk (≈ 0.3–1 MB), and the trained FAME model is only on the order of tens of megabytes, making deployment straightforward. From a complexity standpoint, kernel/GP training requires forming and factoring an $n\times n$ kernel ($O(n^2)$ memory, $O(n^3)$ time) with $O(n)$ per‑test inference, while basis/FPCA costs scale with basis size $p$ (e.g., $O(np^2 + p^3)$), making kernels practical mainly for small $n$ and basis/FPCA expressivity limited by $p$. In contrast, FAME processes each sample once per epoch, so the per‑epoch cost is approximately linear in $n$, with runtime dominated by the NCDE solver and only modest per‑sample overhead is required by attention and MoE in FoFR—roughly $O(T d^2)$ for cross‑attention and $O(TK)$ for routing. Moreover, in typical function‑to‑function regression workloads the grids are short (usually $T<100$) and the number of functions/channels is modest (often $d<20$), so these costs are small in practice. Consequently, investing about an hour to train a fully data‑driven model that delivers substantial accuracy gains, handles irregular and misaligned grids, and preserves seconds‑level inference latency is a practical and worthwhile trade‑off.

Runtime

Memory Usage

• Question (2)：We apologize for the confusion and agree that your suggestion will improve clarity. In the main text, Theorems 1–3 establish Lipschitz properties for different modules. Because concrete module instantiations and computations appear throughout, Appendix A.1 was added to show the end‑to‑end Lipschitz property of the overall framework. Each of Theorems 1–3 corresponds to items (i)–(iii) in Appendix A.1, respectively; however, this mapping was not made explicit. In the revision, we will give (i)–(iii) explicit titles and numbering (e.g., Appendix A.1(i)), and add explicit pointers in the main text so that Theorems 1–3 directly reference their proofs in Appendix A.1(i)–(iii)(e.g., “Proof in Appendix A.1(i)”).

• Question (3) ： Thank you for the suggestion. As shown in Fig. 2, varying the number of sampling points $T\in \{5,10,20,50\}$ yields MSEs $\{0.131,0.086,0.078,0.051\}$. We also tested denser grids with $T=80$ and $T=100$, obtaining MSEs $0.048$ and $0.064$, respectively; these do not change our conclusions and were omitted from Table 1 due to space. We did not pursue substantially longer grids $(T>100)$ for two reasons: first, our task is FoFR, which treats the function as a whole—sampling points are merely a discretization of the underlying curve—whereas long‑sequence interpolation/extrapolation focuses on the state at specific time steps; accordingly, a good FoFR model should be discretization‑invariant, i.e., able to represent functions across different numbers of samples and sampling grids, and the fewer‑point regime is actually the more challenging one. Our method remains accurate and stable for $T\in[5,100]$ (Appendix Figs. 5–6 show that $T\in[20,50]$ already captures the functions well). Second, real‑world acquisitions are typically sparse and irregular, and moving from more data to fewer is straightforward, so demonstrating strong performance with fewer samples is the more informative stress test. In a subsequent version, we will extend Fig. 2(b) to include the $T=80,100$ results for completeness. Finally, from Question (1)’s analysis and the runtime table, our training time scales approximately linearly with $T$, without sharp growth in compute or memory cost, and the method remains lightweight in absolute terms—hence FAME is practically scalable for FoFR.

• Question (4): We fully agree that reporting variability is essential to demonstrate the robustness of our results. In the revision, we will evaluate all methods over 5 random train/test splits × 2 seeds for both synthetic and real‑world tasks and present mean ± std in every table. (Due to time constraints, the table below represents we have currently completed; we can increase the number of runs in a future version if needed.)

  Model	Case 1 (100)	Case 1 (200)	Case 1 (500)	Case 2 (100)	Case 2 (200)	Case 2 (500)
  FAME w/o Bi‑dir	0.1832± 0.0150	0.0812± 0.0066	0.0654± 0.0055	0.1530± 0.0125	0.0528± 0.0023	0.0355± 0.0016
  FAME w/o MoE	0.1870± 0.0161	0.0828± 0.0072	0.0663± 0.0062	0.1578± 0.0132	0.0538± 0.0023	0.0362± 0.0017
  FAME w/o Cross‑attn	0.1902± 0.0158	0.0815± 0.0078	0.0668± 0.0064	0.1602± 0.0143	0.0544± 0.0026	0.0375± 0.0018
  FAME	0.1806± 0.0152	0.0783± 0.0064	0.0635± 0.0056	0.1532± 0.0120	0.0511± 0.0023	0.0342± 0.0017

  Model	Hawaii Ocean Salinity	Hawaii Ocean Temp	Human3.6M Walking	Human3.6M Eating	Human3.6M Sitting down	ETDataset Oil Temp
  Basis Expansion (best)	0.0780 ± 0.0039	0.0014 ± 0.0001	0.0359 ± 0.0018	0.0484 ± 0.0024	0.0122 ± 0.0006	0.0365 ± 0.0018
  FPCA	0.0865 ± 0.0043	0.0025 ± 0.0001	0.0373 ± 0.0019	0.0099 ± 0.0005	0.0121 ± 0.0007	0.0355 ± 0.0018
  Kernel Method	0.0754 ± 0.0038	0.0025 ± 0.0001	0.0373 ± 0.0019	0.0099 ± 0.0005	0.0121 ± 0.0006	0.0355 ± 0.0016
  Gaussian Process	0.0931 ± 0.0047	0.0022 ± 0.0001	0.0360 ± 0.0018	0.0075 ± 0.0004	0.0107 ± 0.0005	0.0380 ± 0.0019
  FNN	0.0766 ± 0.0038	0.0020 ± 0.0001	0.0373 ± 0.0019	0.0099 ± 0.0005	0.0121 ± 0.0006	0.0355 ± 0.0018
  FAME w/o bidir attention	0.0751 ± 0.0023	0.00124 ± 0.00004	0.0327 ± 0.0010	0.0035 ± 0.00010	0.0071 ± 0.0002	0.0264 ± 0.0008
  FAME w/o MoE	0.0759 ± 0.0023	0.00130 ± 0.00004	0.0332 ± 0.0010	0.0038 ± 0.00011	0.0075 ± 0.00023	0.0271 ± 0.0008
  FAME w/o cross‑attention	0.0773 ± 0.0024	0.00145 ± 0.00005	0.0344 ± 0.0011	0.0044 ± 0.00013	0.0083 ± 0.00025	0.0286 ± 0.00086
  FAME	0.0748 ± 0.0022	0.0012 ± 0.00004	0.0325 ± 0.0010	0.0034 ± 0.00010	0.0070 ± 0.00020	0.0262 ± 0.00080

  Once again, thank you for your suggestions and careful review; we hope these additional explanations help you better understand our work and appreciate it.

We thank the reviewer for the helpful suggestions, which will make our work stronger. We have carefully revised the paper to address all comments to the best of our ability. Below we respond point‑by‑point. Due to the submission format and character limitations of the rebuttal, we have focused here on presenting only the newly added or key sections; the formatting and full content will be adjusted as needed for the final manuscript.

• Weaknesses 1:　FoFR is an important problem in practice—yet it remains challenging. We acknowledge that FAME is built from established components (NCDEs, MoE, attention). Our aim, however, was not to refine a particular prior technique nor to “stack” modules for incremental gains, but to design a purpose‑built, fully data‑driven FoFR framework that addresses the core challenges( functions as samples, irregular/misaligned sampling, and continuous outputs ), while preserving global structure and modeling local dynamics. Specifically, our contributions are: (1) a sampling‑invariant continuous attention mechanism that forms continuous Q/K/V via bi‑directional NCDEs，neither this specific approach nor the underlying problem has been proposed in the literature; (2) a shared router with direction‑specific experts so that vector‑field experts specialize to cross‑function heterogeneity without parameter bloat; (3) cross‑channel attention to aggregate inter‑functional dependencies at each time; (4) theoretical support—end‑to‑end Lipschitz stability, sampling invariance, and operator‑level expressivity—grounding the design beyond heuristics; (5) task‑specific design choices for FoFR: the whole function is the sample (motivating bidirectionality), functions differ in shape/amplitude (handled by MoE vector fields), and the output must be a continuous function (realized by an NCDE decoding head). Due to space limitations, we cannot present complete comparison with existing methods here. If interested, please refer to our response to Reviewer 2, where we provide detailed theoretical and experimental comparisons between our method and contemporary Transformer‑based and neural operator approaches.

  In the revised manuscript we will add targeted ablations and hyperparameter analyses to demonstrate the indispensable contribution of each component, substantiating that FAME is not a simple juxtaposition but a principled architecture tailored to FoFR. The detailed results and analyses are reported in our response to Weaknesses  2;

• Weaknesses  2（Question 1）:　We agree that our analysis of the MoE needs clearer exposition. Beyond reporting accuracy, we will quantify routing sparsity and patterns using normalized routing entropy $\tilde H = H/\log K$ . To isolate how $K$ affects both performance and routing behavior under high dimensionality and heterogeneity, we add Case‑8 ($d=10$, RBF widths $\sigma_j\in\{0.2,0.3,0.5\}$) and sweep $K\in\{1,2,3,5,8\}$. The table below presents our experimental results; the final version will also include the corresponding visualizations. In our study, $K=3$ and $K=5$ achieve stable MSE with mid‑range $\tilde H$ (indicating structured, non‑collapsed expert use), while compute cost grows with $K$, so we adopt $K=3$ as the default.

Case‑8: Sensitivity to K

Cross‑attention lets each input function, at each time point, aggregate information from the other functions to capture inter‑functional dependencies, complementing continuous (intra‑function) attention. In the revision, we will report time‑averaged cross‑attention maps and relate attention to function similarity. On Case‑8 (RBF widths $\sigma_j\in\{0.2,0.3,0.5\}$), pairs of functions sharing the same $\sigma_j$ show higher and more similar cross‑attention, forming clusters aligned with the $\sigma_j$ groups. Due to current formatting constraints, we cannot include the figures in this rebuttal; the attention heatmaps and group‑wise summaries will be added in the next revision. To further verify the effectiveness of the attention module, we have also included ablation experiments; because the tables do not fit the rebuttal’s character/format limits, please refer to our response to Reviewer 1 for the complete results.

• Weaknesses  3（Question 2）:　Your concern is fully justified, and we will include quantitative measurements of compute and memory usage. Cases 1–2 jointly span the two principal axes of computational load—the number of functions and the number of sampling points per function—so they are well suited for reporting cost, and all results will be produced on the same ordinary desktop (AMD Ryzen 9 5950X CPU) . In brief, basis/FPCA/kernel methods can train faster when grids are fixed and models are small, but they depend on pre‑specified bases or kernels and may fit particular data poorly; by contrast, FAME is fully data‑driven—while training is more costly, it remains lightweight in absolute terms, and inference time is comparable to classical baselines. Even in our most complex setting, training remains tractable (≈ 207 s/epoch × 20 epochs ≈ 69 min on this CPU), and post‑training inference time is on the order of seconds. Memory usage is even less of a concern and varies little across tasks, so we report average usage: the datasets are purely numeric and small on disk (≈ 0.3–1 MB), and the trained FAME model is only on the order of tens of megabytes, making deployment straightforward. From a complexity standpoint, kernel/GP training requires forming and factoring an $n\times n$ kernel ($O(n^2)$ memory, $O(n^3)$ time) with $O(n)$ per‑test inference, while basis/FPCA costs scale with basis size $p$ (e.g., $O(np^2 + p^3)$); accordingly, classical kernel methods are most practical in the small‑$n$ regime, and basis/FPCA expressivity is ultimately constrained by the chosen basis size $p$—smaller $p$ improves efficiency but limits approximation power; in contrast, FAME processes each sample once per epoch, so the per‑epoch cost is approximately linear in $n$, with runtime dominated by the NCDE solver and only modest per‑sample overhead is required by attention and MoE in FoFR—roughly $O(T d^2)$ for cross‑attention and $O(TK)$ for routing. Moreover, in typical function‑to‑function regression workloads the grids are short (usually $T<100$) and the number of functions is modest (often $d<20$), so these costs are small in practice. Consequently, investing about an hour to train a fully data‑driven model that delivers substantial accuracy gains, handles irregular and misaligned grids, and preserves seconds‑level inference latency is a practical and worthwhile trade‑off.

Runtime

Memory Usage

Once again, thank you for your suggestions and careful review; we hope these additional explanations and experiments help you better understand our work and appreciate it.

We thank the reviewer for the helpful suggestions, which will make our work stronger. We have carefully revised the paper to address all comments to the best of our ability. Below we respond point‑by‑point. Due to the submission format and character limitations of the rebuttal, we have focused here on presenting only the newly added or key sections; the formatting and full content will be adjusted as needed for the final manuscript.

• Question (1): Functional‑on‑Functional Regression (FoFR)—where both inputs and outputs are functions—is a central yet challenging FDA problem that arises in practice, e.g., mapping upstream process signals to thickness/roughness curves in manufacturing or force/torque trajectories to end‑effector motion in robotics. Unlike state‑space time‑series that focus on a single evolving state, here each whole function is one sample. In real industrial settings, each function is often sampled irregularly and sparsely, and different functions are not aligned in their sampling grids. Many deep models implicitly assume fixed, aligned grids or pre‑interpolated sequences, which limits their applicability; consequently, practice often relies on basis‑expansion or kernel methods. Our goal is a fully data‑driven FoFR framework that preserves the global structure emphasized by classical methods while adding local dynamics modeling, with both theoretical guarantees and empirical effectiveness. Different methods’ characteristics on FoFR tasks are summarized in the table below.

  Comparison of FAME and Existing Methods on FoFR Tasks

  Property / Method	LSTM	Transformer	DeepONet	FNO	Basis Expansion	Kernel methods	FAME
  Discretization invariance	✗	✗	✗	✓	~	✓	✓
  Output is a function	✗	✗	✓	✓	✓	✓	✓
  Query output at any point	✗	✗	✓	~	✓	✓	✓
  Take input at any point	~	~	✗	✗	~	✓	✓
  Universal approximation (operator level)	✗	✗	✓	✓	~	✓	✓
  Cross‑resolution train/test	✗	✗	✓	✓	✓	✓	✓
  Data‑driven	✓	✓	✓	✓	✗ (fixed basis)	✗(fixed kernel)	✓

  Discretization invariance means that the model’s output remains essentially unchanged when the same underlying function is sampled on different grids. Universal approximation refers to the model class being able to approximate general function‑to‑function maps. Cross‑resolution train/test indicates that the model can be trained on one sampling density and evaluated on another without retraining or redesign. The symbol “~” denotes partial support via specific preprocessing or variants (e.g., Transformer + Δt), rather than a native property.

  Respecting your suggestion, we will introduce a new Case under the default settings ( $d = 3, \quad m = 1, \quad \Lambda_i = \Gamma_i = 20, \quad N_s = 200, \quad \sigma_j = 0.3$) described in Section 5.1 to compare FAME against LSTM, Transformer, DeepONet, and FNO on both regular‑ and irregular‑sampling FoFR under a unified protocol. To accommodate the baselines, all functions within each setting will share fixed, equal‑length sampling indices. The new experimental results are shown in the table below.

  New Case Results

  Method	Regular‑grid FoFR (MSE )	Irregular‑grid FoFR (MSE )
  LSTM	0.1607±0.0285	0.2180±0.0302
  Transformer	0.0889±0.0202	0.0940±0.0232
  DeepONet	0.1260±0.0076	0.1791±0.0168
  FNO	0.0797±0.0120	0.0944±0.0141
  FAME	0.0775± 0.0062	0.0783± 0.0064

  From the table, FAME remains stable when switching between regular and irregular sampling and stays competitive against other methods. Thus, in both adaptability to diverse scenarios and actual performance, our model shows clear advantages on FoFR.

• Question (2): We fully agree that reporting variability is essential to demonstrate the robustness of our results. In the revision, we will evaluate all methods over 5 random train/test splits × 2 seeds for both synthetic and real‑world tasks and present mean ± std in every table. (Due to time constraints, these represent the experiments we have currently completed; we can increase the number of runs in a future version if needed.)

  Model	Case 1 (100)	Case 1 (200)	Case 1 (500)	Case 2 (100)	Case 2 (200)	Case 2 (500)
  FAME w/o Bi‑dir	0.1832± 0.0150	0.0812± 0.0066	0.0654± 0.0055	0.1530± 0.0125	0.0528± 0.0023	0.0355± 0.0016
  FAME w/o MoE	0.1870± 0.0161	0.0828± 0.0072	0.0663± 0.0062	0.1578± 0.0132	0.0538± 0.0023	0.0362± 0.0017
  FAME w/o Cross‑attn	0.1902± 0.0158	0.0815± 0.0078	0.0668± 0.0064	0.1602± 0.0143	0.0544± 0.0026	0.0375± 0.0018
  FAME	0.1806± 0.0152	0.0783± 0.0064	0.0635± 0.0056	0.1532± 0.0120	0.0511± 0.0023	0.0342± 0.0017

• Question (3): We agree and appreciate this point. In response to the reviewers’ comments, we will add three ablation variants on both generic (Cases 1–2) and real‑data tasks—(1) without bidirectional attention, (2) without MoE (single expert), and (3) without cross‑attention—and report the resulting MSEs side by side to quantify each component’s contribution. The results are shown in the tables provided in our response to Question 2 (Due to space constraints, we regret that the full experimental results on real datasets cannot be shown here; please see our detailed response to Reviewer 1 for the complete results). Note that we do not ablate the continuous‑attention block: because FoFR maps continuous functions to continuous functions, continuous attention is the minimal mechanism that preserves sampling invariance and supports function modeling; removing it would reduce the model to a discrete, grid‑tied surrogate. For further rationale, please see our response to Question 1.

• Question (4): We agree with the concern and will clarify our assumptions and scope. In the main text we stated global Lipschitz/boundedness for brevity, but our results can extend to the standard local‑Lipschitz + linear‑growth setting on a data‑dependent compact set $\mathcal{K}$ that contains the trajectories visited during training/inference. Under this formulation, the NCDE vector fields, cross‑attention map, and decoder each admit local constants $L_{\mathcal{K}}$, yielding the same existence–uniqueness and Grönwall‑type stability bounds with $L_{\mathcal{K}}$ in place of global constants. The end‑to‑end bound then follows by composition of these local constants, and the Rademacher‑style generalization statement becomes explicitly data‑dependent (i.e., it involves $L_{\mathcal{K}}$ and the radius of $\mathcal{K}$ rather than global quantities). We will add a formal Remark and update the Appendix to state all results under local Lipschitz and linear growth.

  High‑dimensional function‑to‑function regression is relatively rare in real applications and highly challenging. Consequently, global constants are often impractical—they may be large or undefined outside $\mathcal{K}$, a risk common to many modern models rather than one specific to ours. What matters in practice is that learned trajectories remain inside a bounded region. This is encouraged in our implementation by input normalization, bounded activations (e.g., $\tanh$ in the vector fields/decoder), softmax with controlled temperature in attention (a Lipschitz map on logits), and optional operator‑norm controls for linear layers (spectral normalization or light‑weight clipping). These standard safeguards keep the effective (local) Lipschitz constants moderate, stabilize the solver, and prevent runaway growth—the regime where our theoretical guarantees apply. We will document these constraints and training guidelines in the revision and explicitly rephrase theorems and assumptions to emphasize local (data‑domain) regularity rather than unrealistic global bounds.

Once again, thank you for your suggestions and careful review; we hope these additional explanations help you better understand our work and appreciate it.

We thank the reviewer for the helpful suggestions, which will make our work stronger. We have carefully revised the paper to address all comments to the best of our ability. Below we respond point‑by‑point. Due to the submission format and character limitations of the rebuttal, we have focused here on presenting only the newly added or key sections; the formatting and full content will be adjusted as needed for the final manuscript.

• How do you decide on the number of experts K? Is this assigned a priori depending on how complex you think the data will be? How does changing K affect results?

  We agree this needs clearer exposition. In a few cases we initially picked $K$ based on the apparent heterogeneity and complexity of the data; however, this choice was preceded by sensitivity checks, and we then selected a value whose performance was stable on a held‑out validation split. We did not present these sensitivity results clearly—sorry for the confusion. In the revision, we will explicitly report this process and add a new synthetic benchmark (Case‑8) to isolate the impact of $K$ under high dimensionality and heterogeneity: the number of input functions is $d=10$; the generating functions use RBF kernels with widths $\sigma_j\in\{0.2,0.3,0.5\}$; and we evaluate $K\in\{1,2,3,5,8\}$. We will report MSE and normalized routing entropy $\tilde H = H / \log K$. The table below presents our experimental results; the final version will also include corresponding visualizations. In our study, $K=3$ and $K=5$ show stable accuracy and healthy expert utilization, while compute cost grows with $K$, so we adopt $K=3$ as the default.

Case‑8: Sensitivity to K

• What are the hyperparameters (e.g. , K, d) used for different real data tests and how did they get selected?

  We denote by d the number of input functions, which equals the number of features in each dataset (see Appendix B.1 for detailed descriptions of d on Hawaii Ocean, Human3.6M and ETT‑small. In our original submission we inadvertently omitted the ETT‑small feature count—it should be d = 6, which will be corrected in the revised version. ). Across our three real‑data benchmarks, d ranges from 2 to 6; based on the sensitivity analysis for K (Question 1), we set K = 3 in all real‑data experiments, since this choice consistently yielded stable accuracy and healthy expert utilization without excessive computational cost. All other architectural and training hyperparameters (e.g. hidden sizes, learning rate, dropout) are described in Appendix B.2. We emphasize that our goal was to demonstrate a general deep‑learning FoFR framework using a relatively simple, unified setup; as a fully extensible architecture, FAME can easily be combined with more complex deep models to suit different datasets.

• Why is the cross attention necessary when there is continuous attention?

  From a theoretical perspective, continuous attention operates within each function over time, performing a continuous, kernelized weighting of its own trajectory to capture local–global intra‑function structure (cf. Eqs. (3)–(5)). Cross‑attention, in contrast, acts across functions at each time t by aggregating the concurrent states of the other functions, thereby modeling inter‑functional dependencies explicitly (cf. Eqs. (7)–(10) and Sec. 4.3). Under Assumption 4, cross‑attention enjoys a global Lipschitz bound $L_{\text{cross}}$ and, when combined with continuous attention and the decoder, yields an end‑to‑end Lipschitz, sampling‑invariant mapping (Theorem 4), so removing cross‑attention weakens the model’s ability to represent cross‑channel couplings.

  From an experimental perspective, and in response to the reviewers’ comments, we will add three ablation variants on both generic (Cases 1–2) and real‑data tasks—(1) without bidirectional attention, (2) without MoE (single expert), and (3) without cross‑attention—and report the resulting MSEs side by side to quantify each component’s contribution. The results will be presented in the table below.

  Model	Case 1 (100)	Case 1 (200)	Case 1 (500)	Case 2 (100)	Case 2 (200)	Case 2 (500)
  FAME w/o Bi‑dir	0.1832± 0.0150	0.0812± 0.0066	0.0654± 0.0055	0.1530± 0.0125	0.0528± 0.0023	0.0355± 0.0016
  FAME w/o MoE	0.1870± 0.0161	0.0828± 0.0072	0.0663± 0.0062	0.1578± 0.0132	0.0538± 0.0023	0.0362± 0.0017
  FAME w/o Cross‑attn	0.1902± 0.0158	0.0815± 0.0078	0.0668± 0.0064	0.1602± 0.0143	0.0544± 0.0026	0.0375± 0.0018
  FAME	0.1806± 0.0152	0.0783± 0.0064	0.0635± 0.0056	0.1532± 0.0120	0.0511± 0.0023	0.0342± 0.0017

  Model	Hawaii Ocean Salinity	Hawaii Ocean Temp	Human3.6M Walking	Human3.6M Eating	Human3.6M Sitting down	ETDataset Oil Temp
  Basis Expansion (best)	0.0780 ± 0.0039	0.0014 ± 0.0001	0.0359 ± 0.0018	0.0484 ± 0.0024	0.0122 ± 0.0006	0.0365 ± 0.0018
  FPCA	0.0865 ± 0.0043	0.0025 ± 0.0001	0.0373 ± 0.0019	0.0099 ± 0.0005	0.0121 ± 0.0007	0.0355 ± 0.0018
  Kernel Method	0.0754 ± 0.0038	0.0025 ± 0.0001	0.0373 ± 0.0019	0.0099 ± 0.0005	0.0121 ± 0.0006	0.0355 ± 0.0016
  Gaussian Process	0.0931 ± 0.0047	0.0022 ± 0.0001	0.0360 ± 0.0018	0.0075 ± 0.0004	0.0107 ± 0.0005	0.0380 ± 0.0019
  FNN	0.0766 ± 0.0038	0.0020 ± 0.0001	0.0373 ± 0.0019	0.0099 ± 0.0005	0.0121 ± 0.0006	0.0355 ± 0.0018
  FAME w/o bidir attention	0.0751 ± 0.0023	0.00124 ± 0.00004	0.0327 ± 0.0010	0.0035 ± 0.00010	0.0071 ± 0.0002	0.0264 ± 0.0008
  FAME w/o MoE	0.0759 ± 0.0023	0.00130 ± 0.00004	0.0332 ± 0.0010	0.0038 ± 0.00011	0.0075 ± 0.00023	0.0271 ± 0.0008
  FAME w/o cross‑attention	0.0773 ± 0.0024	0.00145 ± 0.00005	0.0344 ± 0.0011	0.0044 ± 0.00013	0.0083 ± 0.00025	0.0286 ± 0.00086
  FAME	0.0748 ± 0.0022	0.0012 ± 0.00004	0.0325 ± 0.0010	0.0034 ± 0.00010	0.0070 ± 0.00020	0.0262 ± 0.00080

  As the table shows, compared to the full FAME model, removing cross‑attention degrades performance, highlighting its role in capturing inter‑functional dependencies.

Finally, we again thank you for your careful reading and hope you will enjoy the revised manuscript.