Spectrally-Defined Mollification

• Spectrally-defined mollification is a method that regularizes functions in the spectral domain to enforce boundary compatibility and suppress high-frequency noise.
• It employs convolution with designed spectral kernels and Fourier filters to achieve exponential convergence even for discontinuous or non-smooth data.
• The strategy enhances spectral methods, PINNs, and inverse problem solvers by harmonizing fractional operators with exact boundary conditions for improved accuracy.

A spectrally-defined mollification strategy is an analytical and computational technique that regularizes functions or operator representations by employing mollification in the spectral domain. This approach provides precise boundary compatibility, efficient suppression of high-frequency noise, and exponential convergence properties even for non-smooth or discontinuous data. It is a critical methodology in modern numerical analysis, particularly for spectral methods, inverse problems, and physics-informed neural networks (PINNs) when applied to PDEs involving nonlocal operators and irregular solutions.

1. Spectral Fractional Operators and Boundary Compatibility

Spectrally-defined mollification is intimately connected to the spectral theory of differential operators, most notably the fractional Laplacian on bounded domains. For a smooth domain $\Omega\subset\mathbb{R}^d$ and the Dirichlet Laplacian $-\Delta_D$ with corresponding $L^2(\Omega)$-orthonormal eigenbasis $\{w_j\}_{j\ge1}$ ($- \Delta_D w_j = \lambda_j w_j$, $w_j|_{\partial\Omega} = 0$), the spectral fractional Laplacian is given by

$$\Lambda_D^s f = (-\Delta_D)^{s/2}f = \sum_{j=1}^\infty \lambda_j^{s/2}(f, w_j)_{L^2(\Omega)} w_j$$

for $s \ge 0$, with domain $\mathcal{D}(\Lambda_D^s)$. When $s > 1/2$, $\mathcal{D}(\Lambda_D^s) = H_0^s(\Omega)$, ensuring solutions exhibit exact (“spectral”) Dirichlet boundary conditions (Abdo et al., 4 Jan 2026).

This spectral formulation is essential for PINN solvers of nonlocal PDEs, as standard neural nets do not naturally satisfy boundary compatibility. Mollification in the eigenfunction domain preserves the nonlocal structure and enables rigorous energy estimates in Sobolev spaces.

2. Construction of Spectral Mollifiers

The mollification operator is defined spectrally, either by convolution with specially designed kernels or filtering Fourier or eigenfunction coefficients. On $\Omega$, the spectral mollifier $J_\varepsilon$ is constructed by heat averaging: $$J_\varepsilon \theta(x) = -\frac{1}{\ln\varepsilon}\int_\varepsilon^1 \frac{e^{t\Delta_D}\theta(x)}{t}\,dt = \sum_{j=1}^\infty \rho_j(\varepsilon)(\theta, w_j)_{L^2} w_j(x)$$ with filter weights

$$\rho_j(\varepsilon) = -\frac{1}{\ln\varepsilon}\int_\varepsilon^1 t^{-1} e^{-t\lambda_j}\,dt$$

satisfying $0\le \rho_j(\varepsilon) \le 1$, $\rho_j(\varepsilon)\to1$ as $\varepsilon\to0$, and super-polynomial decay for large $j$ (Abdo et al., 4 Jan 2026).

In Fourier-based spectral methods, as in Gottlieb–Tadmor–Tanner, the mollifier uses entire filters $\varphi_p(\xi)$ of smoothness order $p$ in frequency $\xi$: $$\varphi_{p}(\xi) = \exp(-\tfrac12\xi^2)\sum_{j=0}^p\frac{1}{2^j j!}\xi^{2j}$$ whose inverse Fourier transform yields a Schwartz-class kernel $\Phi_p(y)$. These filters are then localized with width $\delta(x)$ to produce pointwise mollifiers $\Phi_{p,\delta}(x, y)$ tailored to local regularity or proximity to discontinuities (Piotrowska et al., 2017).

3. Spectral Mollification in Numerical Residuals and Regularization

For PDE solvers, spectrally-defined mollifiers are applied to neural network ansatz or spectral reconstructions before forming residuals. In PINN-based approximation of fractional transport-diffusion equations

$$\partial_t\psi + u\cdot\nabla\psi + \Lambda_D^\alpha\psi = f$$

the network output $\psi_\theta$ is mollified by $J_\varepsilon\psi_\theta$, yielding the residual

$$\mathcal{R}_i[\theta](x, t) = \partial_t(J_\varepsilon\psi_\theta) + u\cdot\nabla(J_\varepsilon\psi_\theta) + \Lambda_D^\alpha(J_\varepsilon\psi_\theta) - f$$

which automatically satisfies spectral-boundary compatibility and allows direct energy estimates in Sobolev norms (Abdo et al., 4 Jan 2026).

In severely ill-posed inverse problems, such as Laplace inversion, spectral mollification operates in the log-transformed variable domain: $(Vf)(u) = e^{u/2}f(e^u),\quad u\in\mathbb{R}$ with the mollifier $\varphi_\beta$ acting as a convolution filter. The regularized solution is obtained via variational minimization

$$f_\beta^\delta = \argmin_{f \in L^2(0, \infty)} \left\{ \|g^\delta - Lf\|_{L^2(0, \infty)}^2 + \|(I-C_\beta)Vf\|_{L^2(\mathbb{R})}^2 \right\}$$

achieving stability and optimal convergence rates under natural smoothness assumptions (Maréchal et al., 2023).

4. Rigorous Error Estimates and Convergence Analysis

Spectrally-defined mollification strategies yield rigorous, often exponential, error control. In the PINN context, the error in space-time Sobolev norm

$$\mathcal{E}[\ell, k; \theta]^2 = \int_0^T \sum_{i=0}^k \|\Lambda_D^\ell \partial_t^{(i)}(\psi - J_\varepsilon \psi_\theta)\|_{L^2(\Omega)}^2\,dt$$

is bounded via Grönwall-type inequalities, leveraging mollifier commutation with fractional operators and null boundary leakage: $$\mathcal{E}[\ell, k; \theta]^2 \le C[u]\mathcal{E}_G[\ell+k, k; \theta]^2$$ where $C[u]$ depends on solution regularity and final time (Abdo et al., 4 Jan 2026). Universal approximation results for neural nets, combined with mollification error $\kappa(\varepsilon)$, yield convergence in $H^k$ for arbitrary $k$ as network size $N\to\infty$ and $\varepsilon \to 0$.

In spectral recovery from truncated Fourier series, exponential error bounds hold away from discontinuities: $|f_N^\delta(x) - f(x)| \le C \exp(-\alpha N d(x))$ where $d(x)$ is the distance to the nearest jump, and $N$ is the cutoff frequency (Piotrowska et al., 2017).

For Laplace inversion with mollification, the optimal reconstruction error under Sobolev source conditions is

$$\|f - f_{\beta(\delta)}^\delta\| \le C \rho f_p\left(\frac{\delta^2}{\rho^2}\right)(1 + o(1))$$

where $f_p(t) = [-\ln(t)]^{-p}$, reflecting the logarithmic stability and convergence (Maréchal et al., 2023).

5. Parameter Selection and Practical Implementation

The efficiency of the spectrally-defined mollification strategy relies on adaptive choice of kernel parameters and exact localization. In Fourier methods, the mollifier is adapted pointwise: $$\delta(x) = \sqrt{\theta N d(x)},\quad p = \theta^2 N d(x)$$ with $\theta$ a user constant, $d(x)$ local smoothness radius, and $N$ mode cutoff (Piotrowska et al., 2017). This ensures that smoothing is concentrated away from discontinuities, fully localized near jumps, and only high modes are suppressed elsewhere.

Numerical quadrature cost depends on kernel locality, with naive real-space convolution incurring $O(N^2)$ work, and adaptive quadrature scaling with the Hermite moment order and smooth interval length. Boundary proximity and detected jumps require “one-sided” mollifiers and kernel truncation.

In PDE-based schemes, mollification penalty is imposed directly in coefficient or solution spaces, and standard solvers are used after discrete regularization (Maréchal et al., 2023).

6. Comparative Properties and Limitations

The spectrally-defined approach outperforms pointwise or naive cutoff regularization by guaranteeing commutation with fractional/spectral operators, exact boundary matching, and exponential convergence properties. However, its performance hinges on accurate a-priori or a-posteriori detection of discontinuities or boundaries. Inaccurate edge detection leads to local reduction of convergence exponents. Extension to higher dimensions is nontrivial and requires exploiting tensor structures or fast summation for tractability.

For nonlinear problems, the strategy must be combined with stable time-stepping schemes to avoid amplification of aliased noise in nonlinear terms. Inverse problems such as Laplace inversion remain globally ill-posed but spectrally mollified regularization achieves optimal logarithmic stability (Maréchal et al., 2023).

7. Applications and Broader Impact

Spectrally-defined mollification is foundational in spectral methods for discontinuous data, robust numerical regularization of severely ill-posed inverse problems, and in enforcing spectral boundary conditions in PINNs and neural PDE solvers. It enables both rigorous theoretical analysis and practical, high-accuracy numerical schemes in a wide range of physical and engineering contexts. The strategy has influenced developments in adaptive post-processing, regularization theory, and the design of hybrid neural-spectral algorithms for nonlocal and fractional PDEs (Abdo et al., 4 Jan 2026, Piotrowska et al., 2017, Maréchal et al., 2023).