Physics-Informed TFPONet (PI-TFPONet)

• PI-TFPONet is an unsupervised operator learning framework for parametric PDE interface problems that exploits analytic local solution structures.
• It utilizes tailored finite point operators and local Green’s functions to map discontinuous coefficients and embedded interface conditions efficiently.
• Validated on 1D and 2D high-contrast problems, it achieves relative L2 errors of 10⁻³–10⁻², outperforming supervised methods without relying on labeled data.

Physics-Informed Tailored Finite Point Operator Network (PI-TFPONet) is an unsupervised operator learning architecture devised for parametric interface problems in partial differential equations (PDEs) with discontinuous coefficients and jump data across embedded interfaces. In contrast to conventional Deep Operator Networks (DeepONets), which require copious labeled (input-output) data or physics-informed DeepONets that depend on minimizing stiff PDE-residual terms, PI-TFPONet fully exploits the analytic local solution structure imposed by the governing PDE. This enables the learning of the solution operator for broad families of interface problems—such as those arising in composite media flow, electromagnetics, and material science—without necessitating any precomputed solution data while achieving provable, mesh-uniform convergence, including for singularly perturbed regimes (Du et al., 2024).

1. Problem Setting and Operator Learning Formulation

PI-TFPONet addresses second-order elliptic PDEs with piecewise-smooth coefficients and embedded interfaces. The prototypical model is:

$$\begin{aligned} -\nabla \cdot (a(x)\nabla u(x)) + b(x)u(x) &= f(x) \;\;\; \text{in } \Omega \setminus \Gamma \ [u]_{\Gamma} &= g_D \ [a\nabla u\cdot n]_{\Gamma} &= g_N \ u|_{\partial\Omega} &= 0 \end{aligned}$$

where $\Omega$ is partitioned by $\Gamma$ into subdomains $\{\Omega_k\}$, $a(x)>0$, $b(x)\geq0$ are piecewise smooth, $f$ is variable data, and $[\,\cdot\,]_\Gamma$ denotes the interface jump. The goal is operator learning: approximating the mapping $G: f(\cdot) \mapsto u(\cdot)$ for whole parametric families $(f, a, b, g_D, g_N)$. Unlike supervised methods, no precomputed $(f, u)$ pairs are required. This is critical for problems where high-fidelity simulations are expensive or unavailable (Du et al., 2024).

2. Tailored Finite Point Local Operator Construction

Rather than representing $u(x)$ globally, PI-TFPONet leverages the known local Green's function structure of the PDE on each coarse mesh cell. In 1D, on $[x_{j-1}, x_j]$, with locally constant $a_h, b_h$, the general solution is:

$$u_h(x) = \alpha_j A_1^j(x) + \beta_j A_2^j(x) + \int_{x_{j-1}}^{x_j} F(s) G_j(x, s) ds$$

where $A_1^j, A_2^j$ are fundamental solutions (e.g., exponentials for constant or Airy functions for linearly varying coefficients), $G_j$ is the cell Green's function, and $\alpha_j, \beta_j$ are coefficients. In higher dimensions, e.g., on a quadrilateral $\Delta_j$, the ansatz is:

$$u_h(x, y) = F_0^j/c_0^j + \sum_{k=1}^4 c_{j,k} e^{\pm\mu_j x},\; e^{\pm\mu_j y}$$

with $\mu_j = \sqrt{c_0^j}$, and $F_0^j, c_0^j$ cellwise constants. The mesh size $h$ controls the local approximation. These local formulae encode all available analytic structure, sharply reducing the learning burden to only a handful of local coefficients per cell (Du et al., 2024).

3. Network Architecture

PI-TFPONet inputs a discretized representation of the problem parameters (e.g., $f$, $a$, $b$) and outputs the local solution coefficients per cell:

• 1D: A fully connected network with four layers maps the input vector $[f(x_i)]_{i=1}^M$ to coefficients $\{\alpha_j, \beta_j\}_j$.
• 2D: A convolutional neural network encoder (4 conv layers, latent code dimension 256) processes the input and a decoder (transpose-conv + conv layers) outputs the coefficients $\{c_{j,k}\}$ for each cell.

No additional evaluation network is required for $u_\theta(x)$; reconstruction occurs directly via the analytic local formulae parameterized by the learned coefficients. This separation between physical solution representation and network parameterization is a central innovation (Du et al., 2024).

4. Physics-Informed Unsupervised Loss Formulation

A key distinction is the absence of any bulk PDE-residual loss. Instead, PI-TFPONet enforces compatibility via the following collocation-based, unsupervised loss:

• Continuity Loss: Continuity of both $u_\theta$ and its normal derivative across all interior cell edges.
• Boundary Loss: Enforcement of the prescribed Dirichlet boundary data.
• Interface Loss: Interface jump conditions for both $u_\theta$ and the weighted normal derivative across $\Gamma$.

Explicitly,

$$L(\theta) = \gamma_C \sum_{x\in X^C} \Bigl( [u_\theta]^2 + [\nabla u_\theta \cdot n]^2 \Bigr) + \gamma_B \sum_{x\in X^B} u_\theta(x)^2 + \gamma_J \sum_{x\in X^J} \Bigl( ([u_\theta] - g_D)^2 + ([a\nabla u_\theta\cdot n] - g_N)^2 \Bigr)$$

where $X^C$ (cell interfaces), $X^B$ (boundary), $X^J$ (interface points) denote the appropriate collocation sets and $\gamma_{C,B,J}$ are weights. This loss directly enforces patchwise solution compatibility, eliminating the need to backpropagate through stiff volumetric PDE terms (Du et al., 2024).

5. Theoretical Analysis and Uniform Convergence

Letting $u_\theta(\cdot;f)$ denote the reconstructed solution given input $f$, it holds that

$u_\theta(\cdot;f) = (G_\theta f)(\cdot)$

The main theorem guarantees that if the mesh size $h$ is small and the loss is minimized, then:

$$\|u_\theta - u\|_{2, \Omega}^* \leq C h^2 (\|f\|_{L^2} + \|g_D\|_{\infty, \Gamma} + \|g_N\|_{\infty, \Gamma}) + C \sqrt{L_C(\theta) + L_B(\theta) + L_J(\theta)}$$

for a constant $C$ independent of $h, f, g_D, g_N$. For singularly perturbed problems of the form $-\epsilon \Delta u + cu = f$ (with $\epsilon \ll 1$) the bound holds in an $\epsilon$-weighted norm with $C$ independent of $\epsilon \to 0$. The proof leverages stability estimates for interface problems and quantifies the contribution of both volumetric discretization and interface mismatch residuals (Du et al., 2024).

6. Numerical Validation and Benchmarking

PI-TFPONet was validated on five challenging classes of interface problems:

1. 1D smooth interface,
2. 1D singular perturbation (small $a(x) = \epsilon$),
3. 1D high-contrast $a(x)$ jumps,
4. 2D interface,
5. 2D singular perturbation.

Comparison was made against supervised DeepONet, supervised IONet (interface-aware DeepONet), and their respective physics-informed variants. Key empirical findings include:

• For thin boundary layers and high-contrast coefficients, PI-TFPONet achieved relative $L^2$ errors of $10^{-3}$–$10^{-2}$, outperforming all baselines—including those trained with dense supervision.
• On smooth interface problems, accuracy matched supervised IONet and significantly surpassed vanilla DeepONet or physics-informed DeepONet.
• The network was trained on coarse meshes (32×1 in 1D, 16×16 in 2D), yet generalized robustly to fine-grid (256×1, 128×128) evaluations.
• No labeled data were used at any stage.

A summary table from (Du et al., 2024):

Problem class	PI-TFPONet $L^2$ error	Best baseline $L^2$ error
1D singularly perturbed	$10^{-3}$–$10^{-2}$	$10^{-2}$–$10^{-1}$
1D high-contrast interface	$10^{-3}$–$10^{-2}$	$10^{-2}$–$10^{-1}$
2D interface	Comparable	Comparable

This demonstrates both accuracy and mesh-robust generalization, with the added benefit of label-free, unsupervised operator learning (Du et al., 2024).

7. Contextual Comparison and Innovations

PI-TFPONet diverges fundamentally from prevailing physics-informed operator learning strategies such as PI-DeepONet (Li et al., 18 Aug 2025, Yu et al., 11 Aug 2025), which include the bulk PDE residual in the loss and may suffer from poor conditioning due to stiffness arising from singular perturbations or sharp coefficient jumps. By encoding all analytic solution structure before the network and focusing the loss on glueing local patches and enforcing interface/boundary conditions, PI-TFPONet avoids this pathology and can guarantee mesh-uniform, label-free learning.

The architecture highlights a paradigm shift: analytic locality and physics structure inform the network’s parameterization and training objectives at the outset, reducing the dimension of the learning space and tightly coupling the numerical and machine learning perspectives. This approach is especially beneficial for parametric interface problems where classical label acquisition is computationally prohibitive.

In summary, PI-TFPONet provides a rigorously analyzed, robust, and highly accurate unsupervised operator learning framework for parametric PDEs with interfaces, demonstrating superiority over current data-driven and physics-informed operator networks on challenging interface benchmarks (Du et al., 2024).