Fourier-Enhanced Trunknet DeepONet

• The paper's main contribution is the introduction of a Fourier feature-enhanced trunk that enriches variable representations using a fixed sinusoidal embedding, improving operator learning.
• The architecture processes (z, t, Cv) via a randomized Fourier-feature map before the branch–trunk inner product, leading to enhanced accuracy and computational efficiency compared to standard DeepONet models.
• Quantitative results show that Model 4 achieves lower MSE and faster runtime, effectively capturing rapidly varying and oscillatory function components in consolidation problems.

Searching arXiv for the cited papers and closely related work on Fourier-enhanced trunk networks in DeepONet. The Trunknet Fourier Feature-Enhanced DeepONet is a DeepONet variant in which the trunk network is augmented with a randomized Fourier-feature embedding of the query variables before the standard branch–trunk inner product is formed. In the formulation reported for 1-D consolidation operator learning, this architecture is designated “Model 4” and is introduced to address limitations observed when excess pore-pressure solutions exhibit significant variation. Its defining modification is that the trunk network does not ingest $(z,t,C_v)$ directly; instead, it processes a fixed sinusoidal embedding of these variables, thereby enriching the trunk representation used to construct the operator basis. Within the reported study, this design is the most accurate and the most efficient among the four DeepONet configurations evaluated for the consolidation problem (Choi et al., 14 Jul 2025).

1. Conceptual role within DeepONet operator learning

DeepONet seeks a parametrized operator $\mathcal{O}_\Theta:\mathcal{U}\to\mathcal{S}$ mapping an input function $u(z,0)$ in a function space $\mathcal{U}$ to a solution field $u(z,t)$ in a solution space $\mathcal{S}$. For a query point $y=(z,t)\in D$, the prediction is expressed through a branch–trunk decomposition,

$$\mathcal{O}_\Theta[u](y)\;\approx\;\mathcal{G}(u)(y),$$

with the standard inner-product form

$$\mathcal{O}_\Theta[u](y) \;=\; \sum_{k=1}^q b_k\bigl(u(x_1),\ldots,u(x_m)\bigr)\;t_k(y).$$

In the reported 1-D consolidation setting, the fixed sensor count is $m=100$ and the latent dimension is $\mathcal{O}_\Theta:\mathcal{U}\to\mathcal{S}$0 (Choi et al., 14 Jul 2025).

Within this framework, the branch network encodes the initial excess pore-pressure profile from sensor values, while the trunk network encodes the query variables at which the solution is evaluated. The architectural premise of the trunk Fourier-feature enhancement is that the main representational bottleneck lies in the coordinate encoding. The study first evaluates standard DeepONet variants in which the coefficient of consolidation is embedded in the branch network, then a physics-inspired configuration in which the coefficient is embedded in the trunk network, and finally the Fourier-featured trunk variant. The reported progression indicates that relocating physically salient variables to the trunk already improves performance, and that adding Fourier features further improves the learning of rapidly varying functions (Choi et al., 14 Jul 2025).

A useful contextual distinction is that not all “Fourier-enhanced DeepONet” constructions modify the trunk in the same manner. Some use random Fourier features directly in the trunk input, whereas others replace downstream decoding blocks with Fourier Neural Operator modules or place spectral modules in the branch. The trunknet Fourier feature-enhanced DeepONet refers specifically to the first category.

2. Mathematical formulation and Fourier embedding

The architectural modification is centered on a randomized Fourier-feature map

$\mathcal{O}_\Theta:\mathcal{U}\to\mathcal{S}$1

applied to the trunk input. In Model 4, $\mathcal{O}_\Theta:\mathcal{U}\to\mathcal{S}$2 because the trunk variables are $\mathcal{O}_\Theta:\mathcal{U}\to\mathcal{S}$3, and $\mathcal{O}_\Theta:\mathcal{U}\to\mathcal{S}$4 frequencies are used. The embedding is

$\mathcal{O}_\Theta:\mathcal{U}\to\mathcal{S}$5

where $\mathcal{O}_\Theta:\mathcal{U}\to\mathcal{S}$6 is drawn once from $\mathcal{O}_\Theta:\mathcal{U}\to\mathcal{S}$7 with $\mathcal{O}_\Theta:\mathcal{U}\to\mathcal{S}$8. The resulting embedding dimension is $\mathcal{O}_\Theta:\mathcal{U}\to\mathcal{S}$9, so the trunk input becomes $u(z,0)$0 (Choi et al., 14 Jul 2025).

This construction leaves the branch–trunk output structure unchanged: $u(z,0)$1 What changes is the basis learned by the trunk net. Instead of learning $u(z,0)$2 directly from low-dimensional coordinates, the trunk learns $u(z,0)$3 from a sinusoidal lifting of the coordinates. The study states that this enrichment addresses the identified limitations by capturing rapidly varying functions (Choi et al., 14 Jul 2025).

The rationale is also tied to the analytical structure of the consolidation solution cited in the study,

$u(z,0)$4

The reported interpretation is that embedding $u(z,0)$5 in the trunk is already more aligned with this analytical form than placing it in the branch. The additional Fourier embedding then expands the trunk’s capacity to resolve high-frequency and steep-gradient components, especially at early times. This suggests that the method is not merely a generic positional encoding device, but a targeted modification motivated by the spectral and transient structure of the operator being learned (Choi et al., 14 Jul 2025).

3. Network architecture and representational design

All three MLPs discussed in the study—the branch net, the trunk net, and the merge net where applicable—share the same layer-width pattern: 6 fully-connected hidden layers, each with 30 neurons. The nonlinearity is not explicitly stated as “ReLU” or “tanh”; the study notes that implementations of DeepONet commonly use $u(z,0)$6 activations, and that one may use $u(z,0)$7 or $u(z,0)$8 interchangeably (Choi et al., 14 Jul 2025).

For Model 4, the branch net receives only the sensorized initial condition $u(z,0)$9, so its input dimension is 100. After the 6 hidden layers of width 30, it outputs a $\mathcal{U}$0-dimensional vector. The trunk net also outputs dimension $\mathcal{U}$1, but its input is not the raw coordinate tuple. Instead, the trunk input is the 100-dimensional Fourier embedding $\mathcal{U}$2, followed by the same 6-layer, width-30 MLP stack (Choi et al., 14 Jul 2025).

The architecture can therefore be summarized as follows.

Component	Input	Output
Branch net	$\mathcal{U}$3	$\mathcal{U}$4
Trunk net	$\mathcal{U}$5	$\mathcal{U}$6
Final operator output	branch–trunk inner product	scalar field value

This design is structurally conservative: it preserves the canonical DeepONet factorization and modifies only the representation supplied to the trunk. That is an important technical point, because the reported gains are not attributed to a larger latent dimension or a more elaborate merger mechanism. The improvement is associated with the Fourier lifting of the trunk variables and the resulting ability to learn rapidly varying dependence on $\mathcal{U}$7 (Choi et al., 14 Jul 2025).

4. Training protocol and data generation

The only loss term is the mean-squared error between DeepONet predictions and solver outputs,

$\mathcal{U}$8

where $\mathcal{U}$9 is the number of input functions and $u(z,t)$0 is the number of sampled output locations per function. The reported training set size is 40 000 input functions and the validation set size is 5 000. No additional regularization such as weight decay or PDE-residual terms is applied (Choi et al., 14 Jul 2025).

The data-generation setup uses the spatial domain $u(z,t)$1 and time domain $u(z,t)$2. The coefficient of consolidation is sampled as

$u(z,t)$3

Initial profiles $u(z,t)$4 are either constant in $u(z,t)$5 kPa or sampled from a Gaussian random field with variance $u(z,t)$6 kPa$u(z,t)$7 and length scale $u(z,t)$8. The branch input uses $u(z,t)$9 fixed sensors $\mathcal{S}$0, while each training function is paired with $\mathcal{S}$1 random evaluation locations $\mathcal{S}$2 (Choi et al., 14 Jul 2025).

The training details are reported as not fully specified in the paper, with standard choices listed as Adam, initial learning rate $\mathcal{S}$3, decay when validation loss plateaus, batch size typically 32–256, and training up to 600–1000 epochs, with loss curves shown to 600 epochs (Choi et al., 14 Jul 2025).

A recurrent misconception in the broader neural-operator literature is that “physics-inspired” or “physics-guided” necessarily implies PDE-residual enforcement during optimization. In this model, the only stated objective is supervised MSE against solver outputs. The role of physics enters through variable placement in the architecture and through the analytical interpretation of the solution structure, not through an explicit residual penalty.

5. Quantitative performance and computational characteristics

The reported numerical comparisons show that Model 4 improves both accuracy and runtime relative to the three alternative DeepONet configurations tested in the same study. For the representative case $\mathcal{S}$4 with non-uniform $\mathcal{S}$5, the reported errors are:

Model	Representative MSE (Pa$\mathcal{S}$6)	$\mathcal{S}$7 (Pa)
Model 1	$\mathcal{S}$8	$\mathcal{S}$9
Model 2	$y=(z,t)\in D$0	$y=(z,t)\in D$1
Model 3	$y=(z,t)\in D$2	$y=(z,t)\in D$3
Model 4	$y=(z,t)\in D$4	$y=(z,t)\in D$5

On the test set, the mean $y=(z,t)\in D$6 standard deviation of MSE is reported as $y=(z,t)\in D$7 for Model 1, $y=(z,t)\in D$8 for Model 2, $y=(z,t)\in D$9 for Model 3, and $$\mathcal{O}_\Theta[u](y)\;\approx\;\mathcal{G}(u)(y),$$0 for Model 4. In the ablation, worst-case MSE drops from $$\mathcal{O}_\Theta[u](y)\;\approx\;\mathcal{G}(u)(y),$$1 for Model 3 to $$\mathcal{O}_\Theta[u](y)\;\approx\;\mathcal{G}(u)(y),$$2 for Model 4 (Choi et al., 14 Jul 2025).

The timing results on an NVIDIA RTX 2000 Ada Laptop GPU are also explicit. Computational time per full $$\mathcal{O}_\Theta[u](y)\;\approx\;\mathcal{G}(u)(y),$$3 field is 0.00709 s for Model 1, 0.00822 s for Model 2, 0.00677 s for Model 3, and 0.00619 s for Model 4. The corresponding solver baselines are 0.01118 s for the implicit BDF solver and 0.71304 s for the explicit RK45 solver. The study therefore reports DeepONet models as approximately $$\mathcal{O}_\Theta[u](y)\;\approx\;\mathcal{G}(u)(y),$$4–$$\mathcal{O}_\Theta[u](y)\;\approx\;\mathcal{G}(u)(y),$$5 faster than BDF and approximately $$\mathcal{O}_\Theta[u](y)\;\approx\;\mathcal{G}(u)(y),$$6 faster than RK45, with Model 4 the fastest among the four architectures (Choi et al., 14 Jul 2025).

The study’s interpretation of these results is architectural rather than purely empirical. Model 3 already benefits from placing $$\mathcal{O}_\Theta[u](y)\;\approx\;\mathcal{G}(u)(y),$$7 in the trunk, which the study states reduces MSE by approximately $$\mathcal{O}_\Theta[u](y)\;\approx\;\mathcal{G}(u)(y),$$8 over Model 1. Model 4 then adds Fourier embedding so that high-frequency or steep-gradient components, especially at early times, can be learned more effectively. In worst-case samples, maximum absolute error spikes at early $$\mathcal{O}_\Theta[u](y)\;\approx\;\mathcal{G}(u)(y),$$9 are reported to be reduced by approximately $$\mathcal{O}_\Theta[u](y) \;=\; \sum_{k=1}^q b_k\bigl(u(x_1),\ldots,u(x_m)\bigr)\;t_k(y).$$0 when Fourier features are used. Out-of-distribution tests further indicate that Model 4 remains accurate for $$\mathcal{O}_\Theta[u](y) \;=\; \sum_{k=1}^q b_k\bigl(u(x_1),\ldots,u(x_m)\bigr)\;t_k(y).$$1, degrades gracefully outside that range, and exhibits lower error for smoother initial profiles with correlation lengths $$\mathcal{O}_\Theta[u](y) \;=\; \sum_{k=1}^q b_k\bigl(u(x_1),\ldots,u(x_m)\bigr)\;t_k(y).$$2, while $$\mathcal{O}_\Theta[u](y) \;=\; \sum_{k=1}^q b_k\bigl(u(x_1),\ldots,u(x_m)\bigr)\;t_k(y).$$3 yields higher error but still within reasonable bounds (Choi et al., 14 Jul 2025).

6. Relation to adjacent Fourier-enhanced operator-learning approaches

The trunknet Fourier feature-enhanced DeepONet belongs to a broader line of work that incorporates Fourier structure into operator networks, but the specific locus of Fourier augmentation varies considerably across papers. In the 1-D consolidation study, the Fourier enhancement is an explicit random Fourier-feature map applied to the trunk input $$\mathcal{O}_\Theta[u](y) \;=\; \sum_{k=1}^q b_k\bigl(u(x_1),\ldots,u(x_m)\bigr)\;t_k(y).$$4 before a standard MLP trunk (Choi et al., 14 Jul 2025).

A closely related use of Fourier features appears in a bi-fidelity DeepONet for predicting time trajectories of drag and lift coefficients, where the Fourier feature network is employed as the trunk network because of its proficiency in capturing and learning the oscillatory nature of the target solution with high precision. There, the operator maps the undisturbed free-stream velocity $$\mathcal{O}_\Theta[u](y) \;=\; \sum_{k=1}^q b_k\bigl(u(x_1),\ldots,u(x_m)\bigr)\;t_k(y).$$5 m/s and time $$\mathcal{O}_\Theta[u](y) \;=\; \sum_{k=1}^q b_k\bigl(u(x_1),\ldots,u(x_m)\bigr)\;t_k(y).$$6 s to $$\mathcal{O}_\Theta[u](y) \;=\; \sum_{k=1}^q b_k\bigl(u(x_1),\ldots,u(x_m)\bigr)\;t_k(y).$$7 or $$\mathcal{O}_\Theta[u](y) \;=\; \sum_{k=1}^q b_k\bigl(u(x_1),\ldots,u(x_m)\bigr)\;t_k(y).$$8, and the Fourier trunk is paired with a branch network in the standard DeepONet sum $$\mathcal{O}_\Theta[u](y) \;=\; \sum_{k=1}^q b_k\bigl(u(x_1),\ldots,u(x_m)\bigr)\;t_k(y).$$9. That work also combines the Fourier trunk with a physics-guided sinusoidal parametrization and a two-stage bi-fidelity residual framework (Mollaali et al., 2023).

By contrast, “Fourier-DeepONet” for variable-source full waveform inversion does not explicitly introduce a random Fourier-feature map in the trunk. Instead, it uses a parameter-space DeepONet in which the branch produces a tensor, the trunk encodes source parameters, and a modified Fourier Neural Operator acts as the decoder after branch–trunk merging. The paper states that one may interpret the first Fourier layer as constructing spatial–temporal basis functions, but the architecture is not the same as a trunk-input random Fourier embedding (Zhu et al., 2023).

An even closer conceptual relative is “FEDONet,” which replaces the plain fully connected trunk with a Fourier-embedded trunk using

$m=100$0

and reports average relative $m=100$1 performance gains of $m=100$2–$m=100$3 over a DeepONet baseline across several PDE-driven datasets. The explicit motivation given there is the mitigation of trunk spectral bias in standard MLP trunks (Sojitra et al., 15 Sep 2025). This closely parallels the rationale of Model 4 in the consolidation study, although the latter is framed as an architectural investigation tied to the analytical structure of consolidation solutions.

Related hybrid DeepONet surrogates for multiphase flow also integrate Fourier structure, but primarily in the branch rather than the trunk. In those models, FNO modules encode spatial fields in the branch, while the trunk remains an MLP or KAN handling time and, if desired, spatial query coordinates. This yields spatial–temporal decoupling rather than a Fourier-embedded coordinate trunk (Santos et al., 4 Nov 2025).

Taken together, these works indicate that “Fourier-enhanced DeepONet” is not a single architecture class. It may denote a random Fourier-feature trunk, a Fourier-featured trunk specialized to oscillatory targets, an FNO decoder appended to a DeepONet merger, or an FNO-based branch within a hybrid operator network. The trunknet Fourier feature-enhanced DeepONet is specifically the variant in which the trunk input is lifted by a fixed randomized sinusoidal map to improve the learning of rapidly varying coordinate dependence, and, in the reported 1-D consolidation experiments, that modification yields the best combination of accuracy and efficiency among the tested DeepONet architectures (Choi et al., 14 Jul 2025).