We consider the problem of learning a mapping between function spaces, which can be formalized as the approximation of solution operators for partial differential equations (PDEs). Let ℱ denote a space of input functions, such as initial or boundary conditions, and let 𝒰 denote the corresponding space of solution functions defined over a domain Ω. The goal is to approximate an operator (1)𝒢:ℱ→𝒰,f↦u=𝒢(f),where u satisfies a PDE with input f. For example, in the case of time-dependent problems, f may represent an initial state, while u(x,t) represents the evolution of the system at later times.

From a functional analytic perspective, it is natural to consider u as an element of a Lebesgue space L2(Ω) or, in the presence of derivatives, a Sobolev space H1(Ω). The operator 𝒢 then acts between these Hilbert spaces, and the quality of approximation can be measured using norms such as (2)‖𝒢~(f)−𝒢(f)‖L2(Ω)or‖𝒢~(f)−𝒢(f)‖H1(Ω),where 𝒢~ denotes the learned operator.

In practice, we are given a finite dataset {(f(b),u(b))}b=1B, where u(b)=𝒢(f(b)) is evaluated on a discrete grid. The task of neural operator learning is then to construct a parametric mapping 𝒢~θ that approximates 𝒢 uniformly over ℱ, with the additional constraint that the learned mapping should generalize to unseen inputs and remain invariant to the resolution of the discretization. This formalization provides the foundation for the multi-representation architecture described in Method section.

Let f:Ω→R denote an input function sampled on a fixed grid {xi}i=1N⊂Ω, represented as a vector f=(f(x1),…,f(xN))∈RN.

Learning an operator acting on such functions requires representations that capture both local variations and global structure. To this end, the proposed approach combines three complementary representations of the input function.

First, the discrete pointwise representation of f is processed by a multilayer perceptron, providing a flexible, non-structured embedding capable of approximating arbitrary nonlinear dependencies.

Second, a spectral representation based on Chebyshev polynomials is employed to explicitly encode global properties of the function. For a normalized grid xi∈[−1,1], the Chebyshev coefficients are computed as (3)ck(f)=2N∑i=1Nf(xi)Tk(xi),k=0,…,K−1,where Tk denotes the Chebyshev polynomial of the first kind of degree k. The resulting coefficient vector c(f)∈RK provides a compact description of the global behavior of the function and is further processed by a learnable nonlinear map. More precisely, the spectral embedding hspec(f)∈RH referenced in Eq. (6) is obtained from c(f) through a single fully connected layer followed by layer normalization and a tanh nonlinearity:(4)hspec(f)=tanh(LayerNorm(Wspecc(f)+bspec)),with learnable parameters Wspec∈RH×K and bspec∈RH. This lifts the compact K-dimensional spectral description of f into the same H-dimensional latent space as the pointwise and low-rank branches, so that all three representations can be fused by the gating mechanism introduced below.

Third, a low-rank linear representation is introduced to impose structural constraints on the model. This representation is defined as (5)hlr(f)=tanh⁡(B(Af)),where A∈Rr×N and B∈RH×r with r≪min(N,H). Such a factorization can be viewed as a low-rank approximation of a dense linear operator and serves as an inductive bias favoring global correlations and parameter efficiency. It is worth emphasizing that, although the factorized form W=BA coincides algebraically with the low-rank linear layers used for weight compression in deep networks, its role here is different. In standard compression, a low-rank factorization is applied after training as a surrogate for a dense layer, with the sole purpose of reducing the parameter count. In NOASLRR the low-rank branch is learned from scratch as one of several complementary views of the input function: it is not a compressed replacement for the MLP branch, but an additional branch that is fused with the pointwise and spectral embeddings through the adaptive gates. In this role, the factorization acts as a structural prior that biases the branch towards representing dominant global correlations of f through a small number of latent directions (the rows of A), which complements the high-expressivity pointwise branch and the smooth global description provided by the Chebyshev branch.

The goal is to approximate an operator 𝒢:f↦u(⋅),where u(x) denotes the value of the output function at a query location x∈Rd.

The proposed neural operator follows the DeepONet paradigm and consists of a branch network encoding the input function and a trunk network encoding the spatial coordinate. The operator is defined as (6)𝒢~(f)(x)=S(hmlp(f)⏟pointwise embedding⊙hspec(f)⏟spectral embedding⊙hlr(f)⏟low-rank embedding⊙t(x)⏟trunk),where ⊙ denotes the Hadamard product and S(⋅) is an aggregation operator implemented as a summation over latent components.

Equivalently, the operator can be expressed in compact form as (7)𝒢~(f)(x)=∑i=1Hti(x)himlp(f)hispec(f)hilr(f),which highlights the multiplicative interaction between the latent representations of the input function and the spatial coordinate.

While the multiplicative formulation in Eq. (6) is expressive, directly combining all representations may be unnecessarily restrictive. To address this, adaptive gating mechanisms are introduced.

First, the spectral and low-rank embeddings are combined via a learnable gate:(8)haux(f)=α(f)⊙hspec(f)+(1−α(f))⊙hlr(f),where α(f)=σ(Waux[hspec(f),hlr(f)]+baux),and σ(⋅) denotes the sigmoid function. Here [⋅,⋅] denotes concatenation along the feature dimension, so that [hspec(f),hlr(f)]∈R2H, and Waux∈RH×2H together with baux∈RH are learnable parameters of a single fully connected layer. The sigmoid is applied element-wise, which yields a gating vector α(f)∈(0,1)H that assigns an independent, input-dependent weight to each latent coordinate.

The resulting auxiliary representation is then combined with the pointwise embedding in a residual manner:(9)h(f)=β(f)⊙hmlp(f)+(1−β(f))⊙haux(f).

The second gate β(f)∈(0,1)H is defined analogously to α(f), namely β(f)=σ(Wfin[hmlp(f),haux(f)]+bfin),with learnable parameters Wfin∈RH×2H and bfin∈RH, and is again computed from a single fully connected layer followed by an element-wise sigmoid. Hence the two gates share the same functional form but operate on different pairs of branches.

This adaptive fusion allows the model to balance unstructured expressivity with structured inductive biases depending on the input function.

Using the fused branch representation h(f), the final operator takes the standard DeepONet form (10)𝒢~(f)(x)=∑i=1Hti(x)hi(f).

This formulation preserves the mesh-invariant property of DeepONet while substantially enriching the function representation, as illustrated in Fig. 1.

Theorem 1: Approximation of operators with decomposable structure Let ℱ⊂C([−1,1]) be a compact set of input functions, 𝒳⊂Rd a compact domain, and 𝒢:ℱ×𝒳→R a continuous operator admitting the decomposition𝒢(f,x)=𝒢1(f,x)+𝒢2(f,x), where

𝒢1(f,x)=g1(ℓ1(f),…,ℓm(f),x) depends on f through finitely many linear functionals ℓi:ℱ→R,

Then, for any ε>0, there exists an operator network 𝒩 such thatsupf∈ℱ,x∈𝒳|𝒢(f,x)−𝒩(f,x)|<ε.

Proof: Step 1: Approximation of functions by finite-dimensional subspace.

Since ℱ is compact in C([−1,1]), for any δ>0 there exists a finite-dimensional subspace VK=span{T0,…,TK}

(e.g., Chebyshev polynomials) such that for every f∈ℱ there exists fK=∑k=0Kck(f)Tk∈VKwith‖f−fK‖∞<δ.

Here, ‖x‖∞:=max(|x1|,…,|xn|).

Step 2: Approximation of 𝒢1.

𝒢1(f,x)=g1(ℓ1(f),…,ℓm(f),x) depends on finitely many continuous linear functionals ℓi. Hence |ℓi(f)−ℓi(fK)|≤‖ℓi‖⋅‖f−fK‖∞≤‖ℓi‖δ.

By continuity of g1, |𝒢1(f,x)−𝒢1(fK,x)|≤L1δfor some constant L1>0. Therefore, it suffices to approximate 𝒢1 using the finite set of coefficients ck(f), which can be implemented with the Chebyshev branch and low-rank linear layers.

Step 3: Approximation of 𝒢2.

𝒢2 is continuous on the compact set ℱ×𝒳. By the universal approximation theorem for MLPs, there exists a neural network 𝒩2 such that supf∈ℱ,x∈𝒳|𝒢2(f,x)−𝒩2(f,x)|<δ.

Step 4: Combining the approximations.

Define 𝒩(f,x)=α(f,x)𝒩1(f,x)+(1−α(f,x))𝒩2(f,x),where α(f,x)∈[4] is a gating function (which can itself be approximated by a small MLP). Then |𝒢(f,x)−𝒩(f,x)|≤|𝒢1(f,x)−𝒩1(f,x)|+|𝒢2(f,x)−𝒩2(f,x)|<2δ.

Choosing δ=ε/2 gives the desired ε-approximation.

Step 5: Conclusion.

The Chebyshev branch and LowRank layer approximate 𝒢1, the MLP branch approximates 𝒢2, and the gating layers combine them. Therefore, the Operator network 𝒩 exists with supf∈ℱ,x∈𝒳|𝒢(f,x)−𝒩(f,x)|<ε. ◻

The model is trained in a supervised operator learning setting. Given a dataset 𝒟={(f(b),{(xn(b),yn(b))}n=1Nb)}b=1B,where yn(b)=𝒢(f(b))(xn(b)), the parameters θ are obtained by minimizing the empirical risk (11)ℒ(θ)=1B∑b=1B1Nb∑n=1Nb‖𝒢~θ(f(b))(xn(b))−yn(b)‖2.

The incorporation of spectral and low-rank representations introduces explicit inductive biases that improve convergence and generalization, while the adaptive fusion mechanism ensures sufficient flexibility across diverse operator learning tasks.

Heat Equation

We consider the heat equation on the 2D unit square Ω=[4]2 with prescribed Dirichlet boundary conditions, in which the operator maps the boundary data f (together with a zero initial interior state) to the diffused field u obtained after a fixed number of explicit time-stepping iterations. The governing equation is (12)∂u∂t=α∇2u=α∑i=1n∂2u∂xi2,x∈Ω⊂Rn,t∈[0,T]where α is the thermal diffusivity and f=u|∂Ω encodes the boundary trace. The solution field u is defined on the full 2D grid, which explains the 2D visualizations in Fig. 2; the earlier description of this benchmark as “1D” in the initial submission was a typographic error.

Burgers’ Equation

The Burgers’ equation is defined as (13)∂u∂t+(u⋅∇)u=ν∇2u,x∈Ω⊂Rnwhere ν is the viscosity parameter. 2D visualizations in Fig. 3.

Laplace Equation

The 2D Laplace equation on a rectangular domain Ω=[4]2 is given by (14)∇2u=∑i=1n∂2u∂xi2=0,x∈Ω⊂Rnwhere g(x,y) denotes Dirichlet boundary conditions. 2D visualizations in Fig. 4.

For all problems, we generate datasets {f(b),u(b)}, where f(b) is the input function (initial condition or boundary condition) and u(b) is the corresponding solution on a fixed grid. Each solution is evaluated at a set of query points and used for supervised training with MSE loss:(15)ℒ(θ)=1B∑b=1B1N∑i=1N|𝒢~θ(f(b))(xi)−u(b)(xi)|2.

All three benchmarks use the same architecture and training recipe, which is summarized below and was used to obtain all results reported in this paper.

Datasets. For each PDE we generate 1000 pairs (f(b),u(b)) on a uniform 10×10 grid (N=100 input values per sample), so that f(b)∈R100 encodes the boundary/initial data and u(b)∈R100 is the corresponding discretized solution field. Boundary conditions are generated by drawing four corner values from 𝒩(0,1) and bilinearly interpolating along the edges. The reference solutions are computed with an explicit finite-difference solver for the heat equation (1000 steps, α=0.5, Δt=0.01), with a semi-implicit solver for the viscous Burgers equation (500 steps, ν=0.01, Δt=0.001), and with Jacobi iteration for the Laplace equation (tolerance 10−4, at most 104 iterations). The 1000 samples are split into 80% training and 20% test following the order of generation.

Architecture. The branch and trunk networks both use L=2 hidden layers of width H=128 with layer normalization and tanh activations. The Chebyshev branch uses K=10 modes and the low-rank branch uses rank r=3, so that A∈R3×100 and B∈R128×3. The trunk network takes 2D coordinates as input (R2→R128). Both gates α and β are implemented as a single fully connected layer of shape R256→R128 followed by an element-wise sigmoid.

Training. All models are trained with Adam using learning rate η=3 × 10−4, mini-batches of 32 samples, and Nepochs=500 epochs. The loss is the per-point MSE defined in Eq. (15). Experiments were run in PyTorch on a single GPU; all quantitative results reported below are averaged over 3 independent runs with different random seeds.

We visualize the performance of the proposed model using three types of plots for each test sample. For each benchmark, the figures below show a representative test sample with the input condition, the ground-truth solution, the NOASLRR prediction, and the absolute error; quantitative comparisons with the DeepONet and FNO baselines (including relative L2 and L∞ errors averaged over 3 seeds) are reported in Table 1, so that the visual results below should be read together with that table.

1.

Initial condition or boundary condition (f)

This subsection complements the training curves reported above with additional quantitative evidence requested to make the comparison with DeepONet and FNO more rigorous.

Comparison with Baselines (Mean ± Std)

Table 2 reports the final test MSE, the relative L2 error ‖u−𝒢~(f)‖2/‖u‖2, the relative L∞ error, for DeepONet, FNO and NOASLRR. All error metrics are averaged over 3 independent runs with different random seeds and reported as mean ± standard deviation. Across all three benchmarks NOASLRR achieves the lowest error on every metric while maintaining a compact architecture.

Ablation Study

To isolate the contribution of each branch, we retrain NOASLRR on the heat equation after removing one component at a time: (i) without the Chebyshev spectral branch, (ii) without the low-rank branch, (iii) without the adaptive gates (replaced by an equal-weight average of the three branches), and (iv) with the pointwise MLP branch alone, which recovers a tuned DeepONet. As summarized in Table 3, removing the spectral branch raises the test MSE by roughly two orders of magnitude, confirming that the Chebyshev coefficients are the single most important source of improvement on this smooth benchmark. Removing the low-rank branch also degrades the error significantly, showing that the two structured branches are genuinely complementary rather than redundant. Replacing the gates with a fixed average produces an intermediate but still substantially higher error, confirming the value of input-dependent fusion.

Hyperparameter Sensitivity

The architecture introduces two new hyperparameters relative to DeepONet: the number of Chebyshev modes K and the rank r of the low-rank branch. Table 4 summarizes the sensitivity of NOASLRR on the heat equation when these are varied around the default values (K,r)=(10,3). The model is robust to moderate changes: varying K from 6 to 16 changes the test MSE by less than a factor of two, and varying r from 2 to 8 has a comparably mild effect. Very small K (K≤4) is the only regime in which the Chebyshev branch becomes too coarse to resolve the boundary data, and very large ranks (r≥16) begin to overfit on this small-scale benchmark. The default configuration (K,r)=(10,3) lies in the robust interior of this region.

Gating Analysis

To shed light on how the learned gates behave, we recorded the distribution of α(f) and β(f) on the test set of each PDE. For the heat equation, the average gate values are α≈0.45 and β≈0.41; for the Laplace equation they are α≈0.46 and β≈0.41; and for the Burgers equation α≈0.44 and β≈0.39. The near-equal split between the structured (spectral/low-rank) and unstructured (MLP) branches across all three benchmarks indicates that the model learns to leverage both types of information simultaneously, rather than collapsing to a single branch. The slightly lower β values (relative to α) suggest that the auxiliary representation—combining Chebyshev and low-rank features—receives somewhat more weight in the final fusion than the raw MLP branch, consistent with the benefit of structured inductive biases observed in the ablation study.

Mesh Invariance

Although NOASLRR is trained on a fixed 10×10 grid, its DeepONet-style trunk network makes it possible to evaluate the learned operator at arbitrary query points. To verify that the formulation preserves the claimed mesh invariance, we evaluate the heat-equation model on 20×20 and 40×40 grids obtained by bilinear interpolation of the training inputs and by querying the trunk at the finer coordinates. The relative L2 error at the training resolution (10×10) is 2.6 × 10−2; when evaluating the same trained model on the 20×20 grid the error increases moderately, and on the 40×40 grid it increases further. This gradual degradation—rather than catastrophic failure—confirms that the DeepONet-style trunk formulation provides meaningful generalization across discretizations, although performance at resolutions far from the training grid naturally decreases due to distribution shift in the input representation.

The proposed adaptive multi-representation neural operator demonstrates fast convergence and high accuracy across different PDE benchmarks. As shown in Table 1, the model achieves low mean squared error (MSE) after a relatively small number of training epochs, indicating efficient learning of the underlying operators.

For the 2D heat equation (Table 1), the MSE decreases rapidly from 1.36 at initialization to 1.27×10−4 after 500 epochs, demonstrating that the model accurately captures the evolution of the solution from the initial condition. Similarly, for the viscous Burgers’ equation (Table 1), the model achieves stable convergence, reaching an MSE of 4.62×10−2, despite the nonlinear advection term that typically increases the difficulty of approximation. Finally, the 2D Laplace equation (Table 1) shows that the operator effectively learns the mapping from boundary conditions to the solution field, achieving an MSE below 10−3 after 500 epochs.

These results indicate that the combination of pointwise, spectral, and low-rank representations, together with adaptive gating, enables the operator to efficiently learn both local and global patterns in the solution space. Visual inspections of the predicted solutions confirm that the learned operator reproduces the key structures of the ground truth solutions, and the absolute errors remain low across the domain. Overall, the framework provides both rapid convergence and high-fidelity approximation, demonstrating its potential as a general-purpose operator learning tool.

The training results presented in Fig. 5 demonstrate the effectiveness of the proposed NOASLRR method (shown in yellow) compared to baseline models such as DeepONet and FNO. The loss curves plot all three methods on a shared axis for each equation, so the gap between NOASLRR and the baselines is directly visible; the corresponding final-test statistics are reported in Table 2 and the ablation components in Table 3.

For the Heat Equation, NOASLRR achieves significantly lower MSE loss throughout the training process, converging faster and exhibiting less fluctuation than DeepONet. FNO, while stable, remains at a higher error plateau, indicating that it may struggle to capture certain dynamics of this problem.

Similarly, for the Burgers’ Equation, NOASLRR outperforms both DeepONet and FNO, maintaining lower loss values over 500 epochs. This suggests that our method provides improved approximation capabilities for nonlinear PDEs, benefiting from the tailored architecture and regularization strategy that stabilize training and enhance generalization.

Overall, these results highlight the robustness and efficiency of the proposed NOASLRR approach in learning complex operator mappings, consistently achieving faster convergence and lower errors than the considered baselines.