LNN-PINN: Lie-Encoded Physics-Informed Networks

• LNN-PINN is an advanced PINN architecture that integrates Lie symmetries or Lehmann representations to enforce physical invariants and analytic constraints.
• The framework augments standard loss functions with symmetry-based or Lehmann layers, leading to significant improvements in error metrics and convergence rates.
• LNN-PINN enhances solution fidelity in nonlinear PDEs and quantum impurity problems by inherently enforcing causality and invariant properties without requiring extra penalty tuning.

A Lie-network-encoded Physics-Informed Neural Network (LNN-PINN) refers to an enhanced PINN architecture in which abstract mathematical structures—specifically Lie point symmetries or Lehmann representations—are embedded into the neural network or its loss to enforce invariance or analytic constraints otherwise difficult to guarantee via standard penalty terms. LNN-PINN frameworks have been developed in both the context of partial differential equations (PDEs) and quantum impurity problems to deliver substantial improvements in solution quality, analytic fidelity, and convergence rates by integrating the fundamental mathematical symmetry and analytic structure of the underlying physical systems directly into the learning process (Shah et al., 30 Sep 2025, Kakizawa et al., 2024).

1. Conceptual Foundations

The essential principle behind LNN-PINN is the direct encoding of physical or analytic invariants into the learning process. In the context of PDEs, this manifests as the incorporation of infinitesimal generators from one-parameter Lie groups associated with the PDE operator; for quantum systems, it leverages the Lehmann spectral representation for Green's functions and self-energies. Unlike generic PINNs—which minimize mismatches with data, PDE residuals, and boundary/initial conditions—LNN-PINNs achieve a closer alignment with invariant manifolds or analytic properties intrinsic to the physical model.

In the PDE setting, the Lie symmetry group provides infinitesimal transformations under which the governing equation remains invariant. Embedding these symmetry transformations in the training objective enforces that the neural approximation not only satisfies the equation at nominal collocation points but also along entire symmetry orbits, thereby constraining it to the correct geometric invariance class (Shah et al., 30 Sep 2025).

For many-body quantum systems, the Lehmann representation is a pole-expansion structure for Green's functions and self-energies, automatically enforcing causality, Kramers–Kronig relations, and high-frequency moment constraints. By introducing a dedicated neural "Lehmann layer" and imposing non-negativity and analytic requirements on pole weights, LNN-PINNs can encode these nontrivial physical constraints without resorting to explicit penalty terms or handcrafted loss contributions (Kakizawa et al., 2024).

2. Mathematical Formulation

PDE Case: Lie Symmetry-Enhanced Loss

For PDEs admitting Lie symmetries—such as Burgers' equation—the LNN-PINN augments the standard residual loss with a symmetry-based term constructed from infinitesimal generators $\mathbf{X}$:

$$\mathbf{X} = \xi(x,t,u)\,\partial_x + \tau(x,t,u)\,\partial_t + \eta(x,t,u)\,\partial_u$$

Given a set of collocation points $\{(x_j^c, t_j^c)\}$, their Lie-transformed counterparts are

$$\tilde{x}_j = x_j^c + \varepsilon\,\xi(x_j^c, t_j^c, u_\theta(x_j^c, t_j^c)), \quad \tilde{t}_j = t_j^c + \varepsilon\,\tau(x_j^c, t_j^c, u_\theta(x_j^c, t_j^c))$$

The total loss is

$$\mathcal{L}(\theta) = \alpha\,\mathcal{L}_\text{init} + \beta\,\mathcal{L}_\text{bound} + \gamma\,\mathcal{L}_\text{PDE} + \lambda\,\mathcal{L}_\text{sym}$$

with

$$\mathcal{L}_\text{sym} = \frac{1}{N_c} \sum_j \left| \mathcal{R}[u_\theta](\tilde{x}_j, \tilde{t}_j) \right|^2$$

The symmetry loss enforces that the learned solution is close to being invariant under the admitted symmetries of the PDE (Shah et al., 30 Sep 2025).

Quantum Systems: Lehmann Network PINN

In quantum impurity problems, the LNN-PINN architecture includes a Lehmann layer:

• The neural network maps parameters (e.g., $U$, hybridization function values) to non-negative weights $\{c_p\}$.
• The self-energy is reconstructed as

$$\Sigma(i_k) = \sum_{p=1}^{N_P} \frac{c_p}{i_k - \overline{\omega}_p}$$

where $\overline{\omega}_p$ is a fixed grid in real-frequency space. This form directly mirrors the Lehmann representation of the self-energy and enforces causality and high-frequency default asymptotic structure. The ReLU activation ensures $$\mathbf{X} = \xi(x,t,u)\,\partial_x + \tau(x,t,u)\,\partial_t + \eta(x,t,u)\,\partial_u$$0, guaranteeing the analytic and causal properties in the Matsubara domain.

The loss is pure data-mismatch, as the network architecture inherently enforces all analytic constraints without added regularization:

$$\mathbf{X} = \xi(x,t,u)\,\partial_x + \tau(x,t,u)\,\partial_t + \eta(x,t,u)\,\partial_u$$1

(Kakizawa et al., 2024).

3. Network Architectures and Training Procedures

Lie Symmetry LNN-PINN

• Architecture: Fully connected feed-forward MLP, often eight hidden layers of 40 neurons each.
• Activation: Standard (Tanh, GELU, etc.), with the option for adaptive scale parameters per layer or neuron (in the m-ASPINN variant).
• Initialization: Glorot-normal, fixed random seed.
• Collocation: Typically $$\mathbf{X} = \xi(x,t,u)\,\partial_x + \tau(x,t,u)\,\partial_t + \eta(x,t,u)\,\partial_u$$2 initial/boundary points, $$\mathbf{X} = \xi(x,t,u)\,\partial_x + \tau(x,t,u)\,\partial_t + \eta(x,t,u)\,\partial_u$$3 interior collocation points.
• Optimizer: Adam, with piecewise-constant decay scheduing up to 50,000 iterations.
• Workflow: Each batch computes the standard residual at collocation points, applies Lie generator transformations to compute the symmetry residual at transformed points, and sums losses with prescribed weights.

Lehmann Representation LNN-PINN

• Architecture: Layers 1–4 are dense ReLU with large width (e.g., 1696); layer 5 outputs pole weights, Lehmann layer reconstructs output on Matsubara grid.
• Input: Physical parameters (U, Matsubara hybridization) on an IR grid.
• Output: Real and imaginary parts of self-energy on same grid.
• Loss: Pure data-mismatch, as structural constraints are hard-encoded.
• Training: Adam optimizer, full-batch (batch size = 275), initial learning rate $$\mathbf{X} = \xi(x,t,u)\,\partial_x + \tau(x,t,u)\,\partial_t + \eta(x,t,u)\,\partial_u$$4, reducing midway, for ~125,000 epochs. Training requires several hours on modern GPU hardware (Kakizawa et al., 2024).

4. Theoretical Guarantees and Invariance Properties

The LNN-PINN approach for PDEs is grounded in the infinitesimal criterion of invariance: a differential operator $$\mathbf{X} = \xi(x,t,u)\,\partial_x + \tau(x,t,u)\,\partial_t + \eta(x,t,u)\,\partial_u$$5 is invariant under a Lie group generated by $$\mathbf{X} = \xi(x,t,u)\,\partial_x + \tau(x,t,u)\,\partial_t + \eta(x,t,u)\,\partial_u$$6 if its second prolongation $$\mathbf{X} = \xi(x,t,u)\,\partial_x + \tau(x,t,u)\,\partial_t + \eta(x,t,u)\,\partial_u$$7 vanishes on the solution manifold $$\mathbf{X} = \xi(x,t,u)\,\partial_x + \tau(x,t,u)\,\partial_t + \eta(x,t,u)\,\partial_u$$8. By directly embedding the symmetry residual as an additional loss, the solution is driven toward lying within or near the invariant manifold of interest.

No formal convergence rate improvement or explicit error bound for the combined PINN + Lie symmetry loss has been derived, but numerical evidence across canonical nonlinear PDEs supports substantial increases in solution fidelity (by 1–3 additional digits compared to a standard PINN with the same expressivity and data budget) (Shah et al., 30 Sep 2025).

In the quantum context, the Lehmann layer ensures built-in causality, high-frequency asymptotics, and moment sum-rules, with consistency checks such as the curvature constraint $$\mathbf{X} = \xi(x,t,u)\,\partial_x + \tau(x,t,u)\,\partial_t + \eta(x,t,u)\,\partial_u$$9 for self-energies. This provides strong guarantees that otherwise require careful balancing of explicit penalties in conventional architectures (Kakizawa et al., 2024).

5. Empirical Performance and Quantitative Gains

In the PDE domain, LNN-PINN achieves dramatic reductions in absolute error:

• Burgers' equation ($\{(x_j^c, t_j^c)\}$0, $\{(x_j^c, t_j^c)\}$1):
  • Standard PINN: Pointwise absolute errors $\{(x_j^c, t_j^c)\}$2–$\{(x_j^c, t_j^c)\}$3
  • Lie-symmetry PINN (m-SPINN): Errors $\{(x_j^c, t_j^c)\}$4–$\{(x_j^c, t_j^c)\}$5
  • Symmetry PINN with adaptive activations (m-ASPINN): Errors $\{(x_j^c, t_j^c)\}$6–$\{(x_j^c, t_j^c)\}$7, on par or superior to leading numerical benchmarks (MCB-DQM, WA-DQM, LS-QB-FEM) (Shah et al., 30 Sep 2025).
• Overhead for the symmetry-residual term remains modest; total computational cost is dominated by backpropagation.

In the Lehmann representation context for quantum impurity models:

• Absolute errors $\{(x_j^c, t_j^c)\}$8 across the test grid, including cases with five orders of magnitude separation in self-energy scale.
• The quasiparticle weight $\{(x_j^c, t_j^c)\}$9 is reproduced with high fidelity ($$\tilde{x}_j = x_j^c + \varepsilon\,\xi(x_j^c, t_j^c, u_\theta(x_j^c, t_j^c)), \quad \tilde{t}_j = t_j^c + \varepsilon\,\tau(x_j^c, t_j^c, u_\theta(x_j^c, t_j^c))$$0 almost everywhere).
• Ablation tests demonstrate that omitting the Lehmann layer results in up to eightfold increase in error for derived observables such as electron filling, underscoring the critical role of analytic constraint embedding.

6. Variants and Adaptivity

Adaptations, such as m-SPINN (multiple Lie generator terms) and m-ASPINN (joint Lie symmetry and adaptive activation scaling), provide additional performance gains by allowing for a richer enforcement of invariance and improved gradient flow within very deep or stiff networks. By tuning the Lie-residual weight $$\tilde{x}_j = x_j^c + \varepsilon\,\xi(x_j^c, t_j^c, u_\theta(x_j^c, t_j^c)), \quad \tilde{t}_j = t_j^c + \varepsilon\,\tau(x_j^c, t_j^c, u_\theta(x_j^c, t_j^c))$$1 and leveraging adaptive activation scales, one can tailor the architecture to problem stiffness and desired error tolerance without expensive model redesign (Shah et al., 30 Sep 2025).

A plausible implication is that similar architecture-encoded symmetry (or analytic structure) approaches could be generalized to other domains featuring strong invariants—e.g., conservation laws, gauge invariance, or further spectral sum rules.

7. Broader Implications and Significance

The LNN-PINN approach demonstrates the power of mathematical structure–aware augmentation in physics-informed machine learning: hardwiring symmetries or analytic constraints that are otherwise difficult to encode or even define through generic penalty-based objectives. The method serves as a template for integrating higher-level mathematical properties into neural approximators, leading to improved physical interpretability, rapid convergence, and error reduction without increasing representational complexity or tuning new penalty hyperparameters.

The success of both Lie symmetry–informed PINNs for nonlinear PDEs and Lehmann layer PINNs for quantum many-body systems positions LNN-PINN as a unifying class of methods at the intersection of deep learning, symmetry analysis, and theoretical physics (Shah et al., 30 Sep 2025, Kakizawa et al., 2024).