PITI-DeepONet for Time-Dependent PDEs

• The paper introduces a dual-output architecture that reconstructs the current field and predicts its time derivative to drive classical time-stepping schemes.
• It leverages physics-informed and hybrid training losses to reduce error accumulation, outperforming traditional full rollout and autoregressive methods.
• Residual monitoring through reconstruction discrepancies provides a built-in reliability check, ensuring robust long-term predictions in complex PDE settings.

Searching arXiv for the primary PITI-DeepONet paper and closely related DeepONet/physics-informed operator-learning context. Physics-Informed Time-Integrated Deep Operator Network (PITI-DeepONet) is a neural-operator framework for time-dependent partial differential equations that combines physics-informed operator learning with explicit numerical time integration. Rather than predicting future states directly, it learns the time-derivative operator from the current state and advances the solution with classical time-stepping schemes. The architecture is dual-output: it reconstructs the current field and predicts its time derivative, enabling both long-horizon evolution and residual-based reliability assessment during inference. In the reported benchmark problems, this design is introduced as a response to the limitations of full rollout and autoregressive DeepONet strategies for extended temporal inference (Mandl et al., 7 Aug 2025).

1. Conceptual lineage and problem setting

PITI-DeepONet sits at the intersection of two established lines of work in operator learning. The first is DeepONet, which represents an operator through a branch net that processes a discretized input function and a trunk net that processes the query location, with the operator approximation written as

$$\mathcal{G}_\theta(u)(y)=\sum_{k=1}^q b_k(u(x_1),\ldots,u(x_m))\, t_k(y).$$

This branch–trunk construction was introduced for learning nonlinear operators between function spaces and later extended to parametric PDE solution operators (Wang et al., 2021).

The second line is physics-informed DeepONet, in which automatic differentiation is used to impose PDE residuals and initial/boundary conditions as soft penalties during training. In that formulation, the model can be trained with reduced paired data, and, in some cases, without any paired input–output observations beyond initial or boundary information (Wang et al., 2021). That framework established the broader notion that operator learning can be constrained directly by governing equations rather than solely by trajectory matching.

PITI-DeepONet addresses a more specific difficulty: long-term prediction for time-dependent PDEs. The motivating claim is that traditional full rollout (FR) methods, which predict entire trajectories in one pass, often fail to capture causal dependencies and generalize poorly outside the training time horizon, whereas autoregressive (AR) approaches accumulate error through sequential prediction (Mandl et al., 7 Aug 2025). A closely related development, TI-DeepONet, reformulated the learning target from direct state prediction to the approximation of instantaneous time-derivative fields that are then integrated numerically; PITI-DeepONet inherits this derivative-learning perspective but makes it explicitly physics-informed and adds a dual-output reconstruction mechanism (Nayak et al., 22 May 2025).

2. Dual-output architecture and temporal tangent space operator

The central architectural modification in PITI-DeepONet is that the network does not produce only a future state. Its branch net receives the current field state $\mathbf{u}^n$, while its trunk net receives the spatial coordinates together with “phantom” time zeros indicating the current time. The network then outputs two quantities:

$$\mathcal{G}_{\theta}:\mathbf{u}^n \mapsto (\hat{\mathbf{u}^n}, \widetilde{\mathbf{u}_t^n}).$$

Here, $\hat{\mathbf{u}^n}$ is a reconstruction of the current field, and $\widetilde{\mathbf{u}_t^n}$ is the predicted time derivative at the current state (Mandl et al., 7 Aug 2025).

This is described as learning the temporal tangent space operator: instead of forecasting a future solution manifold point directly, the model approximates the local temporal derivative operator

$$\mathcal{G}_{\theta}: \mathbf{u}^n \mapsto \widetilde{\mathbf{u}_t^n},$$

and then delegates state evolution to a classical integrator (Mandl et al., 7 Aug 2025). The reconstruction output serves two functions. During training, it supports self-consistency constraints. During inference, it supports residual monitoring by comparing the input field to its reconstruction. This design distinguishes PITI-DeepONet from derivative-learning architectures that predict only the right-hand side.

A common misconception is to view PITI-DeepONet as merely another autoregressive rollout model. The architecture is iterative in time, but the learned object is not a next-state map; it is a derivative operator paired with a reconstruction channel (Mandl et al., 7 Aug 2025). That distinction is structurally important because it shifts temporal evolution from a purely learned recurrence to a learned-continuous-dynamics-plus-integrator pipeline.

3. Time integration as the evolution mechanism

At deployment, the current state is passed through the network to obtain $\widetilde{\mathbf{u}_t^n}$, and the state is advanced with a classical time-stepping scheme. The paper reports three such schemes.

With explicit Euler,

$$\mathbf{u}^{n+1}=\mathbf{u}^n+\Delta t\,\widetilde{\mathbf{u}_t^n}.$$

With fourth-order Runge–Kutta (RK4),

$$\begin{aligned} k_1 &= \mathcal{G}_\theta(\mathbf{u}^n),\ k_2 &= \mathcal{G}_\theta\left(\mathbf{u}^n + \tfrac{\Delta t}{2}k_1\right),\ k_3 &= \mathcal{G}_\theta\left(\mathbf{u}^n + \tfrac{\Delta t}{2}k_2\right),\ k_4 &= \mathcal{G}_\theta\left(\mathbf{u}^n + \Delta t\,k_3\right),\ \mathbf{u}^{n+1} &= \mathbf{u}^n + \frac{\Delta t}{6}\left(k_1 + 2k_2 + 2k_3 + k_4\right). \end{aligned}$$

With the Adams–Bashforth–Moulton 2-step predictor–corrector (ABM2),

$$\begin{aligned} \widetilde{\mathbf{u}^{n+1}} &= \mathbf{u}^n + \Delta t\left(\tfrac{3}{2}\mathcal{G}_\theta(\mathbf{u}^n) - \tfrac{1}{2}\mathcal{G}_\theta(\mathbf{u}^{n-1})\right),\ \mathbf{u}^{n+1} &= \mathbf{u}^n + \frac{\Delta t}{2}\left(\mathcal{G}_\theta(\mathbf{u}^n) + \mathcal{G}_\theta(\widetilde{\mathbf{u}^{n+1}})\right). \end{aligned}$$

These formulas make the integration step an explicit part of inference rather than an implicit byproduct of training (Mandl et al., 7 Aug 2025).

This design closely parallels the time-integrated operator-learning idea of TI-DeepONet, where the network approximates the right-hand side and backpropagation proceeds through the integrator (Nayak et al., 22 May 2025). The difference is that PITI-DeepONet couples this with physics-informed objectives and the auxiliary reconstruction output. The reported results further state that Euler, RK4, and ABM2 show near-identical performance on the tested problems, which is presented as evidence that the learned tangent operator is robust to the specific integration scheme chosen at inference (Mandl et al., 7 Aug 2025).

4. Physics-informed and hybrid training objectives

PITI-DeepONet supports fully physics-informed training as well as hybrid physics- and data-driven training. The loss is described as a weighted combination of PDE residual, reconstruction, boundary and initial condition, consistency, and optional supervised field and derivative terms (Mandl et al., 7 Aug 2025).

Two of the explicitly stated terms are the reconstruction loss

$\mathbf{u}^n$0

and the consistency loss

$\mathbf{u}^n$1

where $\mathbf{u}^n$2 denotes the time derivative obtained by automatic differentiation of the reconstructed field $\mathbf{u}^n$3 (Mandl et al., 7 Aug 2025). The consistency term enforces agreement between the network’s explicit derivative output and the derivative implied by the reconstruction channel.

The PDE residual term enforces that $\mathbf{u}^n$4 and its derivatives satisfy the governing equation through automatic differentiation. Boundary and initial condition losses are described as classic physics-informed operator terms, and optional data losses may be added for the field output and the time derivative in hybrid training (Mandl et al., 7 Aug 2025). The weights $\mathbf{u}^n$5 are optimized via hyperparameter search.

This training design clarifies a second common misconception: “physics-informed” here does not imply “data-free only.” The framework is stated to admit both purely physics-informed and hybrid objectives (Mandl et al., 7 Aug 2025). A plausible implication is that PITI-DeepONet is intended not as a rejection of supervised operator learning, but as a way to regularize and stabilize long-horizon evolution whether or not paired data are available.

5. Residual monitoring and inference-time reliability

A distinctive feature of PITI-DeepONet is that it computes a residual during inference from the discrepancy between the input state and its reconstruction:

$\mathbf{u}^n$6

This quantity is used as an intrinsic indicator of reliability and out-of-distribution behavior (Mandl et al., 7 Aug 2025).

The reported interpretation is operationally simple. If the residual is small, the prediction is considered reliable and within the training distribution. If it is large, it signals that the current state lies outside the domain that the model has learned well, warning of possible prediction failure (Mandl et al., 7 Aug 2025). The paper further reports a nearly perfect linear correlation, with Pearson’s $\mathbf{u}^n$7, between this residual and the true squared prediction error for all tested PDEs.

This inference mechanism is not identical to the PDE residual used in training. It is a reconstruction residual, and its role is runtime assessment rather than direct equation enforcement. The paper explicitly presents it as a tool for prediction-quality estimation, out-of-distribution detection, runtime adaptivity, and active learning (Mandl et al., 7 Aug 2025). This suggests that PITI-DeepONet treats operator inference and reliability estimation as coupled tasks rather than separate post hoc procedures.

6. Benchmarks, comparative performance, and relation to adjacent methods

PITI-DeepONet was evaluated on the one-dimensional heat equation, the one-dimensional Burgers equation, and the two-dimensional Allen–Cahn equation. The reported benchmark summary is organized around mean relative $\mathbf{u}^n$8 error reduction relative to full rollout and autoregressive baselines (Mandl et al., 7 Aug 2025).

PDE	Mean error reduction vs. FR	Mean error reduction vs. AR
1D heat equation	84%	79%
1D Burgers equation	87%	98%
2D Allen–Cahn equation	42%	89%

For the heat equation at $\mathbf{u}^n$9, the mean relative $$\mathcal{G}_{\theta}:\mathbf{u}^n \mapsto (\hat{\mathbf{u}^n}, \widetilde{\mathbf{u}_t^n}).$$0 error is reported as $$\mathcal{G}_{\theta}:\mathbf{u}^n \mapsto (\hat{\mathbf{u}^n}, \widetilde{\mathbf{u}_t^n}).$$1 for FR, $$\mathcal{G}_{\theta}:\mathbf{u}^n \mapsto (\hat{\mathbf{u}^n}, \widetilde{\mathbf{u}_t^n}).$$2 for AR, and $$\mathcal{G}_{\theta}:\mathbf{u}^n \mapsto (\hat{\mathbf{u}^n}, \widetilde{\mathbf{u}_t^n}).$$3 for PITI with Euler or RK4/ABM2. For the Burgers equation at $$\mathcal{G}_{\theta}:\mathbf{u}^n \mapsto (\hat{\mathbf{u}^n}, \widetilde{\mathbf{u}_t^n}).$$4, the corresponding means are $$\mathcal{G}_{\theta}:\mathbf{u}^n \mapsto (\hat{\mathbf{u}^n}, \widetilde{\mathbf{u}_t^n}).$$5, $$\mathcal{G}_{\theta}:\mathbf{u}^n \mapsto (\hat{\mathbf{u}^n}, \widetilde{\mathbf{u}_t^n}).$$6, and $$\mathcal{G}_{\theta}:\mathbf{u}^n \mapsto (\hat{\mathbf{u}^n}, \widetilde{\mathbf{u}_t^n}).$$7 or $$\mathcal{G}_{\theta}:\mathbf{u}^n \mapsto (\hat{\mathbf{u}^n}, \widetilde{\mathbf{u}_t^n}).$$8. For the Allen–Cahn equation at $$\mathcal{G}_{\theta}:\mathbf{u}^n \mapsto (\hat{\mathbf{u}^n}, \widetilde{\mathbf{u}_t^n}).$$9, the means are $\hat{\mathbf{u}^n}$0, $\hat{\mathbf{u}^n}$1, and $\hat{\mathbf{u}^n}$2 (Mandl et al., 7 Aug 2025). The abstract characterizes these results as improved accuracy over extended inference time horizons when compared to traditional methods.

Within the broader literature, TI-DeepONet and TI(L)-DeepONet also learn time-derivative fields and integrate them numerically, reporting approximately $\hat{\mathbf{u}^n}$3 error reduction over autoregressive methods and $\hat{\mathbf{u}^n}$4 over fixed-horizon methods, with stable prediction for temporal domains extending to about twice the training interval (Nayak et al., 22 May 2025). PITI-DeepONet differs by combining the derivative-learning and integration strategy with explicitly physics-informed or hybrid losses, a dual-output architecture, and residual-based reliability assessment (Mandl et al., 7 Aug 2025). Relative to earlier physics-informed DeepONets, which established that PDE residuals can regularize operator learning and sometimes remove the need for paired output observations (Wang et al., 2021), PITI-DeepONet specializes that idea for long-term temporal evolution.

The resulting picture is not that PITI-DeepONet replaces all previous operator-learning paradigms, but that it reassigns responsibilities among them: the network learns a temporal tangent operator, the numerical integrator advances the state, and the reconstruction channel monitors trustworthiness. In the reported formulation, this is the key mechanism by which the method moves beyond classic FR and AR schemes for long-time inference of time-dependent PDEs (Mandl et al., 7 Aug 2025).