Physics-Informed Kolmogorov-Arnold Networks

• PIKANs are a hybrid PINN approach that injects sparse or coarse data into the loss function to enhance convergence and mitigate nonconvex challenges.
• They combine physics constraints and data regularization using both simultaneous and two-stage training schemes for robust PDE solutions.
• Empirical results show that minimal high-fidelity data can reduce errors by over an order of magnitude in simulations of complex systems like Navier–Stokes.

Physics-Informed Kolmogorov-Arnold Networks (PIKANs)

Physics-Informed Kolmogorov-Arnold Networks (PIKANs) represent a paradigm in physics-informed neural network (PINN) methodology where sparse, coarse, or experimental data is directly injected into the loss function to regularize the optimization landscape. This hybrid approach leverages both physics (e.g., PDE residuals, boundary, and initial conditions) and data-driven supervision, fundamentally reshaping the loss topology for more favorable convergence properties and mitigating the notorious ill-conditioning and nonconvexity intrinsic to standard PINN training.

1. Combined Loss Function and Theoretical Formulation

Let $u_\theta(X, t)$ denote the PINN output with trainable parameters $\theta$. The loss construction in PIKANs is a weighted sum of physics-informed residuals and data-regulation terms:

$$L_\mathrm{res}(\theta) = L_\mathrm{domain}(\theta) + L_\mathrm{ic}(\theta) + L_\mathrm{bc}(\theta),$$

where

• $$L_\mathrm{domain} = \mathbb{E}_{(X_d, t_d)\in \Omega\times[0,T]} \left[ |\Gamma(u_\theta, t_d) + \Lambda(u_\theta, X_d)|^2 \right]$$,
• $$L_\mathrm{ic} = \mathbb{E}_{X_i\in\Omega} \left[ |u_\theta(X_i, 0) - f(X_i)|^2 \right]$$,
• $$L_\mathrm{bc} = \mathbb{E}_{(X_b, t_b)\in \partial\Omega\times[0,T]} \left[ |u_\theta(X_b, t_b) - g(X_b, t_b)|^2 \right]$$.

The data-regularization term incorporates measured data $\hat{u}$ at sparse inputs $(X_s, t_s)$: $$L_\mathrm{data}(\theta) = \mathbb{E}_{(X_s, t_s)} \left[ |u_\theta(X_s, t_s) - \hat{u}(X_s, t_s)|^2 \right].$$ The total loss is

$$L_\mathrm{total}(\theta) = L_\mathrm{res}(\theta) + \lambda L_\mathrm{data}(\theta),$$

with $\lambda>0$ governing the strength of data regularization relative to physics (residual) loss.

This construction enables a hybrid unsupervised-supervised regime, exploiting the global structure of the physics loss while sharpening the convergence path via targeted empirical data (Gopakumar et al., 2022).

2. Reshaping the Loss Landscape: Mechanisms and Effects

Classical PINNs enforce physics solely via PDE residuals, typically yielding loss surfaces with wide, flat plateaus and multiple shallow minima due to competing domain, initial, and boundary conditions. Such landscapes are highly nonconvex and saddle-laden, often trapping gradient-based optimizers in sub-optimal basins.

Introducing a small proportion of high-fidelity (even sparse) data points "melts" these flat minima and induces sharp gradients around the ground-truth solution. The data loss term acts as a puncture in the loss surface, enforcing local curvature that aligns gradient directions toward the correct basin. This effect persists with coarse or inexact data, provided $\lambda$ is appropriately reduced to prevent bias toward measurement noise.

Visualization of loss landscapes (e.g., 1D Burgers, 2D Wave, 2D Navier–Stokes) confirms a transformation from bumpy/flattish minima under pure physics regularization to narrow, deep valleys with the addition of data-regularization (Gopakumar et al., 2022).

3. Training Procedures: Simultaneous and Two-Stage Hybridization

Two primary training schemes are established:

• Simultaneous Training:
  • At each epoch, draw collocation points for $L_\mathrm{res}$ and data points for $L_\mathrm{data}$.
  • Compute $$\nabla_\theta (L_\mathrm{res} + \lambda L_\mathrm{data})$$ via automatic differentiation.
  • Update $$\theta_{k+1} \leftarrow \theta_k - \eta_k \nabla_\theta (L_\mathrm{res} + \lambda L_\mathrm{data})$$.
• Two-Stage (Curriculum) Training:
  • Stage I (Unsupervised warm-up): Minimize $L_\mathrm{res}$ for $N_1$ epochs.
  • Stage II (Supervised fine-tuning): Minimize $L_\mathrm{res} + \lambda L_\mathrm{data}$ for $N_2$ epochs.
  • Optional: Anneal $\lambda$ towards zero over Stage II so the final model is dominated by physics-consistency, avoiding overfitting to noisy or coarse data.

Both procedures leverage the data term as a regularizer to guide optimization out of suboptimal basins, with annealing mitigating the risk of compromising the hard physical constraints at convergence.

4. Empirical Results and Benchmark Performance

PIKANs yield significant accuracy improvements and faster convergence across PDE benchmarks:

Problem	Vanilla PINN $L_2$ error	+1% Sparse Data	+Coarse-Grid Data
1D Burgers ($\nu=0.01/\pi$)	$7.8\times 10^{-2}$	$1.0\times 10^{-3}$	--
2D Wave Equation	$5\times 10^{-4}$	$3\times 10^{-4}$	--
2D Navier–Stokes (Re=50)	$8.24$	$0.233$	$0.2135$

The presence of just $1\%$ sparse regulatory data can reduce error by more than an order of magnitude and shrinks the solution space toward the unique, correct PDE-constrained profile (Gopakumar et al., 2022).

5. Practical Guidelines for Weighting and Data Selection

Effective implementation depends critically on sensible choices of $\lambda$ and regulatory data density:

• $1$–$5\%$ data relative to collocation points is typically sufficient for strong effect.
• Tune $\lambda$ such that initial magnitudes of $L_\mathrm{res}$ and $\lambda L_\mathrm{data}$ are comparable; monitor their ratio in early epochs.
• For coarse/noisy data, decrease $\lambda$ (e.g., $0.1$ or less).
• If annealing, reduce $\lambda \to 0$ after the first half of training so that the model finally enforces the pure PDE.
• Always check that residuals remain small; excessive weighting of $L_\mathrm{data}$ may undermine hard physics constraints.
• No additional mesh refinement is necessary; data-regularization achieves accuracy without increased computational cost.

6. Comparison to Alternative PINN Regularizations

The data-regularization strategy in PIKANs is complementary to other loss engineering approaches targeting optimization pathologies in PINNs: variance-based regularizers for outlier control (Hanna et al., 2024), dual (Lagrange-multiplier) formulations for high-order PDEs (Basir, 2022), and adversarial or embedded subdomain/curriculum protocols for strong nonlinearity (Krishnapriyan et al., 2021). However, PIKAN data-regularization is unique in its capacity to leverage even highly incomplete data for sharp transformation of nonconvex loss surfaces.

7. Outlook and Impact

PIKANs exemplify a powerful principle: injecting even minimal empirical data can exert disproportionate influence on the convergence properties and error rates of physics-constrained neural PDE surrogates. This data-informed reshaping of the optimization landscape is critical in advancing PINN methodology for challenging forward and inverse problems where pure enforcement of physics proves insufficiently robust or tractable (Gopakumar et al., 2022).