The Latent Solution: A Finite Sufficient Representation Framework for Partial Differential Equations
Abstract
We introduce the Latent solution as a quantitative framework for finite-dimensional representation of PDE solutions. The framework builds on the Galerkin method but exploits the grade structure of analytic PDEs (Nagy, 2026) to provide four structural guarantees absent from standard Galerkin theory.
For an evolution PDE \(\partial_t u = F(u)\) with analytic data and analyticity radius \(\rho > 1\), the Latent solution is the trajectory \(\Lambda(t) = (c_0(t), \ldots, c_{N^}(t))\) in the truncated spectral coefficient space, where \(N^ = \lceil \log(C_0/\varepsilon) / \log \rho \rceil\) modes are determined by the PDE's own analyticity radius. We prove:
(I) Self-determined truncation (Theorem 1). The analyticity radius \(\rho\) of the PDE data uniquely determines the truncation level \(N^(\varepsilon, \rho)\) with certified error bound \(\|u - u_{N^}\|_{L^2} \leq \varepsilon\).
(II) Grade-structured coupling (Theorem 2). For a PDE whose right-hand side is a polynomial of degree \(K\) in \(u\), the Latent ODE on \(\mathbb{R}^{N^+1}\) has at most \(K\)-body couplings. High-mode self-interactions are exponentially suppressed by the grade bound, giving effective sparsity \(O((N^)^K)\) instead of \((N^)^{N^}\).
(III) Exponential closure (Theorem 3). The closure error of the truncated system — the effect of unresolved modes (\(k > N^\)) on the resolved modes (\(k \leq N^\)) — is bounded by \(O(\varepsilon)\). The truncated system is a certified approximation of the full dynamics, not an uncontrolled projection.
(IV) Spectral interpolation (Theorem 4). The framework interpolates continuously between three regimes:
| Data regularity | Coefficient decay | \(N^*\) | Solution quality | |:---|:---|:---|:---| | Analytic (strip width \(\sigma > 0\)) | \(|c_k| \leq C e^{-\sigma|k|}\) | \(O(\log 1/\varepsilon)\) | Classical, analytic | | \(C^s\) (finite smoothness) | \(|c_k| \leq C |k|^{-s}\) | \(O(\varepsilon^{-1/s})\) | Classical, \(C^s\) | | \(L^2\) (no smoothness) | No guaranteed decay | \(O(\varepsilon^{-d})\) | Weak (standard Galerkin) |
The transition from exponential to algebraic compression at the analyticity boundary is a phase transition in the solution's representability.
Novelty
Repackages classical Galerkin truncation with analyticity-based spectral decay into a unified 'Latent solution' framework that makes the self-determined truncation level N* and its certified error bound explicit, but the individual ingredients (spectral Galerkin, exponential coefficient decay for analytic data, grade-structured coupling sparsity) are well-established in approximation theory and dynamical systems.