Physics-Informed Extreme Learning Machine

• PIELM is a physics-informed solver that integrates a fixed single-hidden-layer network with PDE, boundary, and initial constraints to compute outputs via a linear least-squares system.
• It replaces iterative gradient descent with a convex linear solve using the Moore–Penrose pseudoinverse, achieving significant speedups over traditional PINNs.
• PIELM has been extended across domains like finance, geotechnics, and acoustics, while addressing challenges such as feature conditioning and noise sensitivity.

Physics-Informed Extreme Learning Machine (PIELM), also written PI-ELM, is a family of physics-informed solvers in which the unknown field is represented by a single-hidden-layer feedforward network with fixed hidden-layer parameters and output weights obtained from a linear least-squares system assembled from PDE, boundary, initial, and sometimes data constraints (Dwivedi et al., 2019, Rout, 8 Aug 2025). Originating as a rapid alternative to Physics-Informed Neural Networks (PINNs) for stationary and time-dependent linear PDEs, PIELM preserves the collocation-based, mesh-free, residual-enforcement logic of PINNs but replaces iterative gradient descent with a Moore–Penrose pseudoinverse or equivalent linear solve (Dwivedi et al., 2019). Subsequent work has generalized the framework to inverse problems, uncertainty-aware inference, stiff and high-frequency equations, time-stepping, eigenvalue problems, option-pricing PDEs, geotechnical systems, inverse Stefan problems, and acoustic field interpolation (Liu et al., 2022, Yang et al., 10 Jun 2025, Mishra et al., 21 Aug 2025, Srinivasan et al., 5 Oct 2025, Zhuang et al., 24 Oct 2025, Komaba et al., 7 Jun 2026).

1. Origins, scope, and defining characteristics

The original PIELM formulation was introduced as a rapid method for the numerical solution of stationary and time-dependent linear PDEs, with a distributed extension, DPIELM, proposed for large or complex domains and sharp solutions (Dwivedi et al., 2019). In the later literature, the terms “PIELM” and “PI-ELM” are used interchangeably for the same core idea: combining an Extreme Learning Machine (ELM) with physics-informed residual constraints (Rout, 8 Aug 2025).

The ELM component is a single-hidden-layer network. In one common form, the approximation is written

$$\hat{u}(x)=\sum_{i=1}^{N^*} c_i\,\phi_i(x), \qquad \phi_i(x)=\tanh(m_i^\top x+b_i),$$

with hidden weights $m_i$ and biases $b_i$ drawn once and frozen, and only the output coefficients $c_i$ trained (Srinivasan et al., 5 Oct 2025). Equivalent formulations appear throughout the literature as

$$u_\beta(\mathbf{x})=\phi(\mathbf{W}\mathbf{x}+\mathbf{b})\beta,$$

or, in RBF-based variants,

$$\hat{u}(x)=\sum_{j=1}^{N} B_j \exp\!\left(-\frac{\|x-x_j\|_2^2}{2 s_j^2}\right),$$

again with fixed hidden-layer parameters and trainable output weights only (Rout, 8 Aug 2025, Srinivasan et al., 6 Mar 2026).

The “physics-informed” component means that the network is not trained solely against labeled solution values. Instead, the governing equation, boundary conditions, initial conditions, interface conditions, or auxiliary measurements are imposed at collocation points and become linear constraints on the output weights (Dwivedi et al., 2019, Yang et al., 10 Jun 2025). This design makes PIELM a fixed-feature, physics-constrained solver rather than a deep, fully trainable neural PDE model. This suggests that PIELM should be regarded not merely as a faster optimizer for PINNs, but as a distinct approximation family with its own conditioning, expressivity, and sampling issues.

2. Mathematical formulation and convex training structure

A standard PIELM setup starts from a PDE such as

$$\mathcal{L}(u)(x)+f(x)=0,\quad x\in\Omega, \qquad \mathcal{B}(u)(x)=g(x),\quad x\in\partial\Omega,$$

or, in operator notation used elsewhere,

$$\mathcal{N}[u](\mathbf{x})=0,\qquad u(\mathbf{X}_u)=\mathbf{u}_o$$

(Srinivasan et al., 5 Oct 2025, Rout, 8 Aug 2025). Replacing $u$ by $\hat u$, one defines interior and boundary residuals at collocation points. Because $m_i$0 is linear in the output weights, and because the operator is linear in the solution in the classical PIELM setting, the residuals remain linear in those weights: $m_i$1 Stacking all interior, boundary, and initial equations yields a linear system

$m_i$2

typically overdetermined and solved in the least-squares sense by a pseudoinverse or related linear solver (Srinivasan et al., 5 Oct 2025, Rout, 8 Aug 2025).

This structure makes the training problem convex in the output weights. One paper states the Hessian with respect to $m_i$3 as $m_i$4, hence positive semidefinite, and therefore identifies PI-ELM as a “convex variant of neural PDE solvers” (Rout, 8 Aug 2025). In practice, derivatives needed by the PDE operator are obtained either by automatic differentiation of the hidden units or, in some implementations with simple activations such as $m_i$5, by symbolic differentiation of closed-form expressions (Srinivasan et al., 5 Oct 2025, Yang et al., 10 Jun 2025).

Regularization is not essential to the definition, but several formulations note that Tikhonov or ridge-type regularization can be inserted if the linear system is ill-conditioned: $m_i$6 The Bayesian extension places a Gaussian prior directly on the output weights and converts the least-squares solve into Bayesian linear regression, yielding both posterior means and posterior covariances for forward and inverse linear PDE problems with noisy data (Liu et al., 2022).

3. Relation to PINNs and computational profile

A PINN typically uses a deep, fully trainable network $m_i$7 and minimizes a weighted residual loss by iterative optimizers such as Adam or L-BFGS. PIELM replaces that nonconvex optimization with a single solve for the output layer, while also restricting the architecture to a shallow fixed-feature model (Srinivasan et al., 5 Oct 2025, Rout, 8 Aug 2025). The practical effect is that runtime is dominated by matrix assembly and linear algebra rather than epochs, learning-rate schedules, or optimizer pathologies.

The financial PDE study provides a direct quantitative comparison. For a Black–Scholes European call problem, the reported PINN baseline gives MSE $m_i$8 with runtime $m_i$9 s, whereas tanh-PIELM gives MSE $b_i$0 with runtime $b_i$1 s, a speedup of about $b_i$2 (Srinivasan et al., 5 Oct 2025). For the Heston–Hull–White model, the same study reports PINN MSE $b_i$3 and runtime $b_i$4 s versus tanh-PIELM MSE $b_i$5 and runtime $b_i$6 s, about $b_i$7 faster (Srinivasan et al., 5 Oct 2025). In the soil-consolidation setting, TS-PIELM is reported to improve computational efficiency and accuracy by more than $b_i$8 times and $b_i$9 times, respectively, for one-dimensional cases relative to PINN (Yang et al., 10 Jun 2025). For inverse Stefan problems, the PIELM framework is reported to increase prediction accuracy by $c_i$0-$c_i$1 order of magnitude in terms of relative $c_i$2 error while saving more than $c_i$3 training time relative to conventional PINNs (Zhuang et al., 24 Oct 2025).

These comparisons also clarify a common misconception. PIELM is not simply “PINN without backpropagation.” It changes the optimization landscape, the parameterization, and the failure modes. PINNs are primarily limited by nonconvex training and spectral bias; PIELMs largely eliminate the first issue, but become highly sensitive to feature conditioning, hidden-layer design, and collocation geometry (Rout, 8 Aug 2025).

4. Conditioning, spectral content, and adaptive feature design

A major line of PIELM research argues that approximation failures often stem less from shallow-network expressivity than from the conditioning of the activation matrix. In a 1D steady advection–diffusion benchmark with $c_i$4 hidden nodes and $c_i$5 collocation points, the reported activation matrix $c_i$6 has observed rank $c_i$7, condition number $c_i$8, and determinant approximately zero (Rout, 8 Aug 2025). Under such circumstances, pseudoinverse-based training becomes numerically fragile even though the formal optimization is convex.

Several variants address this bottleneck by redesigning the hidden features while preserving the linear solve. Shifted Gaussian Encoding (SGE) modifies the feature map to

$c_i$9

which localizes neuron responses and enriches the activation matrix. For the same advection–diffusion problem, the reported rank increases from $$u_\beta(\mathbf{x})=\phi(\mathbf{W}\mathbf{x}+\mathbf{b})\beta,$$0 to $$u_\beta(\mathbf{x})=\phi(\mathbf{W}\mathbf{x}+\mathbf{b})\beta,$$1, and the solvable range of $$u_\beta(\mathbf{x})=\phi(\mathbf{W}\mathbf{x}+\mathbf{b})\beta,$$2 improves from approximately $$u_\beta(\mathbf{x})=\phi(\mathbf{W}\mathbf{x}+\mathbf{b})\beta,$$3 to approximately $$u_\beta(\mathbf{x})=\phi(\mathbf{W}\mathbf{x}+\mathbf{b})\beta,$$4 (Rout, 8 Aug 2025).

High-frequency and variable-frequency PDEs motivate a second branch of variants. General Fourier Feature PIELM replaces the usual hidden activation by neuron-wise cosine features,

$$u_\beta(\mathbf{x})=\phi(\mathbf{W}\mathbf{x}+\mathbf{b})\beta,$$5

with $$u_\beta(\mathbf{x})=\phi(\mathbf{W}\mathbf{x}+\mathbf{b})\beta,$$6 distributed across a prescribed frequency interval and refined by inspecting the output-weight distribution $$u_\beta(\mathbf{x})=\phi(\mathbf{W}\mathbf{x}+\mathbf{b})\beta,$$7 (Ren et al., 14 Oct 2025). The same paper reports substantial accuracy gains on variable-frequency wave equations, multi-frequency wave equations, high-frequency Helmholtz problems on irregular domains, Klein–Gordon equations, and advection–diffusion examples, without additional cost in training time or architecture complexity (Ren et al., 14 Oct 2025). Frequency Shift PIELM addresses the same regime by shifting the mean of the Gaussian weight distribution rather than scaling it, and reports that the linear variant attains the best accuracy in six of seven benchmark problems, with improvements of one to nearly five orders of magnitude over existing PIELM variants (Xiong et al., 2 Jul 2026).

Sharp-gradient problems have also driven kernel-adaptive approaches. KAPI-ELM learns a small set of hyperparameters defining the statistical distribution from which RBF centers and widths are drawn, using Bayesian optimization over distributional parameters instead of backpropagation over each hidden weight (Dwivedi et al., 14 Jul 2025). GMM-PIELM treats the residual field as a probabilistic “location of physics,” fits a Gaussian mixture to that residual density, and resamples kernels accordingly; for $$u_\beta(\mathbf{x})=\phi(\mathbf{W}\mathbf{x}+\mathbf{b})\beta,$$8 convection–diffusion, it reports $$u_\beta(\mathbf{x})=\phi(\mathbf{W}\mathbf{x}+\mathbf{b})\beta,$$9 errors up to $$\hat{u}(x)=\sum_{j=1}^{N} B_j \exp\!\left(-\frac{\|x-x_j\|_2^2}{2 s_j^2}\right),$$0 orders of magnitude lower than baseline RBF-PIELM while retaining the speed advantage of the ELM architecture (Srinivasan et al., 6 Mar 2026).

5. Major application domains and domain-specific variants

In quantitative finance, PIELM has been used as a mesh-free solver for option-pricing PDEs. One study formulates both Black–Scholes and Heston–Hull–White pricing as forward problems, and also embeds PIELM inside an outer Bayesian optimization loop for inverse calibration of volatility and interest-rate parameters from noisy option prices (Srinivasan et al., 5 Oct 2025). In that calibration setting, the reported relative errors in $$\hat{u}(x)=\sum_{j=1}^{N} B_j \exp\!\left(-\frac{\|x-x_j\|_2^2}{2 s_j^2}\right),$$1 and $$\hat{u}(x)=\sum_{j=1}^{N} B_j \exp\!\left(-\frac{\|x-x_j\|_2^2}{2 s_j^2}\right),$$2 are below $$\hat{u}(x)=\sum_{j=1}^{N} B_j \exp\!\left(-\frac{\|x-x_j\|_2^2}{2 s_j^2}\right),$$3 on synthetic data (Srinivasan et al., 5 Oct 2025).

In computational geotechnics, TS-PIELM discretizes time and trains a separate ELM at each time interval using a Crank–Nicolson residual, thereby addressing sharp temporal gradients that standard PIELM handles poorly (Yang et al., 10 Jun 2025). A distinct geotechnical application uses PIELM for tunnelling-induced soil-pile interactions, where the pile is modeled as an Euler–Bernoulli beam and the soil as a Pasternak foundation, yielding a fourth-order ODE as the physics-informed component and measured deflection data as the data-driven component. That framework is reported to train within $$\hat{u}(x)=\sum_{j=1}^{N} B_j \exp\!\left(-\frac{\|x-x_j\|_2^2}{2 s_j^2}\right),$$4 second by least squares, and its parametric study concludes that monitored data should be placed where pile-deflection gradients are significant, such as at the pile tip or top and near tunneling zones (Guo et al., 1 Oct 2025).

Phase-change problems constitute another active area. For inverse Stefan problems with known moving interfaces, PIELM replaces the conventional deep PINN by an ELM whose output weights are solved analytically from PDE, initial, boundary, and interface residuals, transforming the inverse problem into a Moore–Penrose generalized inverse computation (Zhuang et al., 24 Oct 2025). In acoustics, a hybrid framework uses PINN pre-training to learn hidden-layer weights for the one-dimensional wave equation and then adapts only the output layer by a PIELM solve for each new target sound field; under simplified 1D free-field conditions, the adaptation time is reported to be reduced by more than three orders of magnitude while retaining interpolation accuracy comparable to PINN-based fine-tuning (Komaba et al., 7 Jun 2026).

PIELM has also been extended beyond standard forward PDE solution. Bayesian PIELM introduces a Gaussian prior over the output layer and yields predictive uncertainty for both forward and inverse linear PDE problems with noisy data (Liu et al., 2022). Eig-PIELM reformulates linear eigenvalue problems into a generalized algebraic eigenproblem, with boundary conditions enforced exactly by algebraic projection onto a boundary-admissible subspace; in the reported beam and acoustic-cavity benchmarks, both eigenvalues and mode shapes are obtained in a single solve (Mishra et al., 21 Aug 2025).

6. Limitations, misconceptions, and current directions

The core one-shot PIELM construction depends on linearity. In the original and many subsequent formulations, the PDE operator must be linear in the unknown field so that residuals remain linear in the output coefficients (Dwivedi et al., 2019, Srinivasan et al., 5 Oct 2025). Nonlinear PDEs break this property, and several papers therefore treat nonlinearity through reformulation, iterative linearization, or curriculum strategies rather than through the original direct solve (Rout, 8 Aug 2025). Time-stepping and domain decomposition serve similar roles: TS-PIELM addresses sharp temporal gradients by breaking the time axis into small intervals, while DPIELM addresses large domains and localized sharp structures by using local PIELMs coupled across interfaces (Yang et al., 10 Jun 2025, Dwivedi et al., 2019).

A second limitation is that PIELM does not eliminate hyperparameter sensitivity; it relocates it. Hidden-layer randomness, basis type, neuron count, width or frequency range, collocation density, and linear-solver regularization all materially affect accuracy. The conditioning studies on stiff advection–diffusion and multi-frequency regression make this point explicit: poor feature geometry can reduce an apparently wide hidden layer to a very low-rank numerical basis (Rout, 8 Aug 2025). Likewise, the high-frequency literature shows that uncontrolled spectral distributions in the hidden layer can be more damaging than insufficient network width (Ren et al., 14 Oct 2025, Xiong et al., 2 Jul 2026).

A third limitation is noise sensitivity. Classical pseudoinverse-based PIELM can overfit noisy data and provides no native uncertainty quantification. Bayesian PIELM remedies this by replacing the point estimate with a posterior over output weights and, in inverse problems, over augmented PDE parameters, thereby yielding predictive variances as well as means (Liu et al., 2022). Acoustic interpolation with a pre-trained PIELM also reports sensitivity to low signal-to-noise ratios, which points to the importance of regularization and noise-aware linear solvers in practical deployments (Komaba et al., 7 Jun 2026).

Finally, there is an important interpretive caution. Because PIELM often achieves large speedups over PINNs, it is sometimes presented as a straightforward acceleration of physics-informed learning. The broader literature suggests a more precise characterization: PIELM is a family of shallow, fixed-feature, physics-informed collocation methods whose success depends on matching the hidden feature distribution to the operator, geometry, and solution spectrum (Rout, 8 Aug 2025, Dwivedi et al., 14 Jul 2025). Current research therefore concentrates less on changing the least-squares core than on changing where the basis lives: in time-stepped coordinates, in subdomains, in adaptive kernel distributions, in Fourier bands, or in boundary-admissible subspaces.