TENG-BC: Unified Neural PDE Solver

• TENG-BC is a unified time-evolving natural-gradient method that intrinsically handles general boundary conditions for neural PDE solvers.
• It combines explicit time integration with a single least-squares formulation to jointly enforce interior dynamics and diverse boundary constraints.
• Benchmarks demonstrate high accuracy over long time horizons, outperforming PINNs and matching fine-mesh FEM in various PDE scenarios.

Searching arXiv for TENG-BC and related TENG neural PDE solver papers. TENG-BC, short for "Unified Time-Evolving Natural Gradient for Neural PDE Solvers with General Boundary Conditions," is a neural method for solving time-dependent partial differential equations by combining explicit time integration with a boundary-aware natural-gradient update (Jiang et al., 28 Feb 2026). In the reported formulation, each time step jointly enforces interior dynamics and boundary conditions, accommodating Dirichlet, Neumann, Robin, and mixed constraints within a unified framework. The method is positioned as a high-precision, mesh-free solver for settings in which long-time error accumulation and boundary enforcement are the dominant difficulties. A closely related precursor, "TENG++: Time-Evolving Natural Gradient for Solving PDEs With Deep Neural Nets under General Boundary Conditions" (He et al., 13 Dec 2025), extended the original Time-Evolving Natural Gradient framework from periodic to Dirichlet boundary conditions by combining natural-gradient optimization with explicit Euler and Heun schemes.

1. Origins, scope, and nomenclature

The defining idea behind the Time-Evolving Natural Gradient family is to replace a vanilla gradient step,

$$\theta_{n+1}=\theta_n-\eta\,\nabla_\theta L(\theta_n),$$

with a geometry-aware update,

$$\theta_{n+1}=\theta_n-\eta\,F(\theta_n)^{-1}\nabla_\theta L(\theta_n),$$

where $F(\theta)$ is a Fisher-information or generalized Gauss–Newton matrix. In the account given for TENG++, the original TENG framework was demonstrated under periodic boundary conditions, while TENG++ extended it to Dirichlet boundary conditions through a loss of the form $$L(\theta)=L_{\rm PDE}(\theta)+\lambda L_{\rm BC}(\theta)$$ and explicit time stepping (He et al., 13 Dec 2025). TENG-BC further reformulates boundary handling so that interior and boundary constraints enter a single local least-squares problem, and the resulting update is interpreted directly as a natural-gradient step under an $L^2(\Omega)$-type metric augmented by boundary contributions (Jiang et al., 28 Feb 2026).

This progression suggests a shift from penalty-based boundary enforcement toward an intrinsic operator-aware update. In TENG++, the Dirichlet condition is imposed by a boundary penalty weighted by $\lambda$; in TENG-BC, the boundary operator appears directly in the metric and residual construction, and the abstract explicitly states that the method operates “without delicate penalty tuning” (Jiang et al., 28 Feb 2026).

The acronym should also be distinguished from an unrelated usage in power-electronics literature, where TENG denotes a triboelectric nanogenerator rather than a Time-Evolving Natural Gradient method (Pathak et al., 2021). In the present context, TENG-BC belongs to neural PDE solvers rather than energy-harvesting circuitry.

2. Continuous formulation and natural-gradient structure

In TENG-BC, the PDE is written as an evolution equation

$$u_t(x,t)=L\,u(x,t),\qquad x\in\Omega,\quad t\in[0,T],$$

subject to a general mixed boundary condition

$a(x)\,u(x,t)+b(x)\,\partial_n u(x,t)=v(x,t).$

A neural network $u_\theta(x)$ with parameters $\theta\approx\theta(t)$ represents the solution at each time step (Jiang et al., 28 Feb 2026).

At a step of size $$\theta_{n+1}=\theta_n-\eta\,F(\theta_n)^{-1}\nabla_\theta L(\theta_n),$$0, the method forms a target field $$\theta_{n+1}=\theta_n-\eta\,F(\theta_n)^{-1}\nabla_\theta L(\theta_n),$$1 determined by the chosen time integrator. The forward Euler target is

$$\theta_{n+1}=\theta_n-\eta\,F(\theta_n)^{-1}\nabla_\theta L(\theta_n),$$2

and higher-order methods, including Heun and RK4, define $$\theta_{n+1}=\theta_n-\eta\,F(\theta_n)^{-1}\nabla_\theta L(\theta_n),$$3 through intermediate evaluations of $$\theta_{n+1}=\theta_n-\eta\,F(\theta_n)^{-1}\nabla_\theta L(\theta_n),$$4 (Jiang et al., 28 Feb 2026). The network is then linearized around $$\theta_{n+1}=\theta_n-\eta\,F(\theta_n)^{-1}\nabla_\theta L(\theta_n),$$5, and one seeks an increment $$\theta_{n+1}=\theta_n-\eta\,F(\theta_n)^{-1}\nabla_\theta L(\theta_n),$$6 such that the induced variation $$\theta_{n+1}=\theta_n-\eta\,F(\theta_n)^{-1}\nabla_\theta L(\theta_n),$$7 approximates the target increment $$\theta_{n+1}=\theta_n-\eta\,F(\theta_n)^{-1}\nabla_\theta L(\theta_n),$$8 in $$\theta_{n+1}=\theta_n-\eta\,F(\theta_n)^{-1}\nabla_\theta L(\theta_n),$$9:

$F(\theta)$0

where $F(\theta)$1.

The normal equations are

$F(\theta)$2

with

$F(\theta)$3

Hence

$F(\theta)$4

The paper identifies this update as “precisely the natural-gradient step under the $F(\theta)$5 metric on function space” (Jiang et al., 28 Feb 2026).

This formulation places the method closer to a local function-space projection than to ordinary first-order optimization. A plausible implication is that the geometry of the network enters only through the Jacobian and the induced metric $F(\theta)$6, which helps explain why the method is reported to maintain low stepwise error over long time horizons.

3. Unified treatment of boundary conditions

The distinctive feature of TENG-BC is its boundary-aware least-squares step. For a general boundary condition $F(\theta)$7 on $F(\theta)$8, the mismatch is defined by

$F(\theta)$9

with boundary Jacobians

$$L(\theta)=L_{\rm PDE}(\theta)+\lambda L_{\rm BC}(\theta)$$0

The boundary-only update solves

$$L(\theta)=L_{\rm PDE}(\theta)+\lambda L_{\rm BC}(\theta)$$1

TENG-BC combines the interior and boundary objectives into a single local least-squares problem over the closure $$L(\theta)=L_{\rm PDE}(\theta)+\lambda L_{\rm BC}(\theta)$$2:

$$L(\theta)=L_{\rm PDE}(\theta)+\lambda L_{\rm BC}(\theta)$$3

or equivalently

$$L(\theta)=L_{\rm PDE}(\theta)+\lambda L_{\rm BC}(\theta)$$4

where $$L(\theta)=L_{\rm PDE}(\theta)+\lambda L_{\rm BC}(\theta)$$5 and $$L(\theta)=L_{\rm PDE}(\theta)+\lambda L_{\rm BC}(\theta)$$6 (Jiang et al., 28 Feb 2026).

The corresponding metric becomes

$$L(\theta)=L_{\rm PDE}(\theta)+\lambda L_{\rm BC}(\theta)$$7

and the gradient is

$$L(\theta)=L_{\rm PDE}(\theta)+\lambda L_{\rm BC}(\theta)$$8

The update remains

$$L(\theta)=L_{\rm PDE}(\theta)+\lambda L_{\rm BC}(\theta)$$9

Boundary type	Parameters
Dirichlet	$L^2(\Omega)$0
Neumann	$L^2(\Omega)$1
Robin	$L^2(\Omega)$2 on $L^2(\Omega)$3
Mixed	Spatially varying $L^2(\Omega)$4 on different boundary segments

This construction differs materially from the Dirichlet-only penalty approach used in TENG++, where

$L^2(\Omega)$5

with

$L^2(\Omega)$6

and

$L^2(\Omega)$7

TENG++ therefore balances PDE and boundary enforcement through a scalar weight $L^2(\Omega)$8, whereas TENG-BC incorporates the boundary operator directly into the least-squares metric (He et al., 13 Dec 2025).

4. Time integration, linear algebra, and implementation

At the algorithmic level, TENG-BC receives current parameters $L^2(\Omega)$9, a time increment $\lambda$0, the operator $\lambda$1, boundary data $\lambda$2, and the network $\lambda$3. It computes $\lambda$4 by the selected time integrator, then performs a small number of least-squares iterations. In each iteration it samples interior points $\lambda$5 and boundary points $\lambda$6, computes

$\lambda$7

and

$\lambda$8

forms the Jacobians, assembles $\lambda$9 and $$u_t(x,t)=L\,u(x,t),\qquad x\in\Omega,\quad t\in[0,T],$$0, solves

$$u_t(x,t)=L\,u(x,t),\qquad x\in\Omega,\quad t\in[0,T],$$1

by, for example, truncated SVD, and updates $$u_t(x,t)=L\,u(x,t),\qquad x\in\Omega,\quad t\in[0,T],$$2 (Jiang et al., 28 Feb 2026).

The practical details reported for TENG-BC are specific. Sampling uses fixed uniform grids of $$u_t(x,t)=L\,u(x,t),\qquad x\in\Omega,\quad t\in[0,T],$$3 interior and $$u_t(x,t)=L\,u(x,t),\qquad x\in\Omega,\quad t\in[0,T],$$4 boundary points, with no resampling needed. Only a subset of parameters, for example 512, is active in each least-squares solve to improve conditioning, and the subset may be re-selected each step. Stability is supported by small singular-value truncation in the least-squares solver and double-precision arithmetic. The experiments use $$u_t(x,t)=L\,u(x,t),\qquad x\in\Omega,\quad t\in[0,T],$$5. Complexity is described as follows: assembling $$u_t(x,t)=L\,u(x,t),\qquad x\in\Omega,\quad t\in[0,T],$$6 costs

$$u_t(x,t)=L\,u(x,t),\qquad x\in\Omega,\quad t\in[0,T],$$7

while the least-squares solve via truncated SVD or QR costs

$$u_t(x,t)=L\,u(x,t),\qquad x\in\Omega,\quad t\in[0,T],$$8

Overall, the method scales linearly in the number of samples (Jiang et al., 28 Feb 2026).

The precursor TENG++ presents the same sequential-in-time philosophy in a more classical predictor-corrector language. Its Euler variant forms

$$u_t(x,t)=L\,u(x,t),\qquad x\in\Omega,\quad t\in[0,T],$$9

then fits $a(x)\,u(x,t)+b(x)\,\partial_n u(x,t)=v(x,t).$0 together with the boundary condition by a few natural-gradient steps. Its Heun variant computes a predictor, evaluates the PDE residual at the predicted state, forms

$a(x)\,u(x,t)+b(x)\,\partial_n u(x,t)=v(x,t).$1

and then performs a second natural-gradient fit (He et al., 13 Dec 2025). TENG++ states that Euler is cheap but first-order accurate in $a(x)\,u(x,t)+b(x)\,\partial_n u(x,t)=v(x,t).$2, whereas Heun attains second-order accuracy while doubling the number of PDE-residual evaluations per time step.

5. Benchmarks and quantitative performance

The benchmark suite for TENG-BC spans diffusion, transport, and nonlinear PDEs under varied boundary conditions (Jiang et al., 28 Feb 2026). For the heat equation $a(x)\,u(x,t)+b(x)\,\partial_n u(x,t)=v(x,t).$3 on the unit disk with $a(x)\,u(x,t)+b(x)\,\partial_n u(x,t)=v(x,t).$4, the tested boundary conditions include inhomogeneous Dirichlet, zero and nonzero Neumann, Robin $a(x)\,u(x,t)+b(x)\,\partial_n u(x,t)=v(x,t).$5 with $a(x)\,u(x,t)+b(x)\,\partial_n u(x,t)=v(x,t).$6 chosen via Bessel identities, and mixed Neumann–Dirichlet on a quarter-annulus. The transport equation

$a(x)\,u(x,t)+b(x)\,\partial_n u(x,t)=v(x,t).$7

is posed on the unit disk with inflow Dirichlet boundary and velocity

$a(x)\,u(x,t)+b(x)\,\partial_n u(x,t)=v(x,t).$8

The viscous Burgers equation

$a(x)\,u(x,t)+b(x)\,\partial_n u(x,t)=v(x,t).$9

is tested on $u_\theta(x)$0 with $u_\theta(x)$1 and periodic boundaries (Jiang et al., 28 Feb 2026).

The reported quantitative results are strong. For the heat equation, the full-time relative $u_\theta(x)$2 error at $u_\theta(x)$3 is approximately $u_\theta(x)$4 for TENG-Euler, approximately $u_\theta(x)$5 for TENG-Heun, and approximately $u_\theta(x)$6–$u_\theta(x)$7 for TENG-RK4. For the transport equation, the per-step error remains $u_\theta(x)$8. For Burgers, relative $u_\theta(x)$9 error is measured against spectral-1024:

Method	$\theta\approx\theta(t)$0	$\theta\approx\theta(t)$1	$\theta\approx\theta(t)$2	$\theta\approx\theta(t)$3
TENG-Euler	$\theta\approx\theta(t)$4	$\theta\approx\theta(t)$5	$\theta\approx\theta(t)$6	$\theta\approx\theta(t)$7
TENG-Heun	$\theta\approx\theta(t)$8	$\theta\approx\theta(t)$9	$$\theta_{n+1}=\theta_n-\eta\,F(\theta_n)^{-1}\nabla_\theta L(\theta_n),$$00	$$\theta_{n+1}=\theta_n-\eta\,F(\theta_n)^{-1}\nabla_\theta L(\theta_n),$$01
TENG-RK4	$$\theta_{n+1}=\theta_n-\eta\,F(\theta_n)^{-1}\nabla_\theta L(\theta_n),$$02	$$\theta_{n+1}=\theta_n-\eta\,F(\theta_n)^{-1}\nabla_\theta L(\theta_n),$$03	$$\theta_{n+1}=\theta_n-\eta\,F(\theta_n)^{-1}\nabla_\theta L(\theta_n),$$04	$$\theta_{n+1}=\theta_n-\eta\,F(\theta_n)^{-1}\nabla_\theta L(\theta_n),$$05
FEM	$$\theta_{n+1}=\theta_n-\eta\,F(\theta_n)^{-1}\nabla_\theta L(\theta_n),$$06	$$\theta_{n+1}=\theta_n-\eta\,F(\theta_n)^{-1}\nabla_\theta L(\theta_n),$$07	$$\theta_{n+1}=\theta_n-\eta\,F(\theta_n)^{-1}\nabla_\theta L(\theta_n),$$08	$$\theta_{n+1}=\theta_n-\eta\,F(\theta_n)^{-1}\nabla_\theta L(\theta_n),$$09
PINN-BFGS	$$\theta_{n+1}=\theta_n-\eta\,F(\theta_n)^{-1}\nabla_\theta L(\theta_n),$$10	$$\theta_{n+1}=\theta_n-\eta\,F(\theta_n)^{-1}\nabla_\theta L(\theta_n),$$11	$$\theta_{n+1}=\theta_n-\eta\,F(\theta_n)^{-1}\nabla_\theta L(\theta_n),$$12	$$\theta_{n+1}=\theta_n-\eta\,F(\theta_n)^{-1}\nabla_\theta L(\theta_n),$$13

The paper summarizes these results by stating that TENG-BC variants maintain uniformly low error over time, outperform PINNs, and match or exceed fine-mesh FEM under comparable sampling budgets. It further characterizes the observed accuracy as “solver-level,” with errors at or below $$\theta_{n+1}=\theta_n-\eta\,F(\theta_n)^{-1}\nabla_\theta L(\theta_n),$$14 for diffusion, approximately $$\theta_{n+1}=\theta_n-\eta\,F(\theta_n)^{-1}\nabla_\theta L(\theta_n),$$15 for advection, and $$\theta_{n+1}=\theta_n-\eta\,F(\theta_n)^{-1}\nabla_\theta L(\theta_n),$$16 near shocks in Burgers (Jiang et al., 28 Feb 2026).

The precursor TENG++ reports a more restricted but informative heat-equation study on the unit disk $$\theta_{n+1}=\theta_n-\eta\,F(\theta_n)^{-1}\nabla_\theta L(\theta_n),$$17 with

$$\theta_{n+1}=\theta_n-\eta\,F(\theta_n)^{-1}\nabla_\theta L(\theta_n),$$18

and initial data given as a linear combination of Bessel-mode eigenfunctions. There, analytical solutions from Bessel expansions enable exact error computation. The stated findings are that TENG-Euler with step-size $$\theta_{n+1}=\theta_n-\eta\,F(\theta_n)^{-1}\nabla_\theta L(\theta_n),$$19 yields errors on the order of $$\theta_{n+1}=\theta_n-\eta\,F(\theta_n)^{-1}\nabla_\theta L(\theta_n),$$20 by $$\theta_{n+1}=\theta_n-\eta\,F(\theta_n)^{-1}\nabla_\theta L(\theta_n),$$21, while TENG-Heun with the same $$\theta_{n+1}=\theta_n-\eta\,F(\theta_n)^{-1}\nabla_\theta L(\theta_n),$$22 achieves errors below $$\theta_{n+1}=\theta_n-\eta\,F(\theta_n)^{-1}\nabla_\theta L(\theta_n),$$23 up to $$\theta_{n+1}=\theta_n-\eta\,F(\theta_n)^{-1}\nabla_\theta L(\theta_n),$$24; Euler is competitive for low to moderate accuracy demands, whereas Heun strongly outperforms for high precision (He et al., 13 Dec 2025).

6. Limitations, extensions, and interpretation

The limitations stated for TENG-BC are primarily linear-algebraic rather than conceptual. Least-squares solves become larger for very high-dimensional networks. The paper reports that the partial-update trick mitigates this issue, but also notes that further scalable solvers, such as randomized sketching, may be needed for extremely high-dimensional PDEs (Jiang et al., 28 Feb 2026). This suggests that the principal bottleneck is not the PDE formulation itself, but the cost and conditioning of repeatedly solving local Jacobian-based least-squares problems.

Several extensions are identified explicitly. TENG-BC lists extension to Cauchy-type and more general boundary-value problems, scalable approximations of the metric $$\theta_{n+1}=\theta_n-\eta\,F(\theta_n)^{-1}\nabla_\theta L(\theta_n),$$25 such as low-rank or block-diagonal forms, incorporation of data assimilation or parametric variations for operator learning, applications to coupled multi-physics or stochastic PDEs, and automatic selection of active-parameter subsets (Jiang et al., 28 Feb 2026). TENG++ likewise describes natural extensions to Neumann conditions through an additional penalty

$$\theta_{n+1}=\theta_n-\eta\,F(\theta_n)^{-1}\nabla_\theta L(\theta_n),$$26

to mixed boundary conditions by combining Dirichlet and Neumann terms on complementary segments, and to nonlinear, coupled, or higher-order PDEs by replacing the diffusion operator with a general $$\theta_{n+1}=\theta_n-\eta\,F(\theta_n)^{-1}\nabla_\theta L(\theta_n),$$27; it also emphasizes that the networks remain mesh-free on complex geometries (He et al., 13 Dec 2025).

A recurrent misconception in this area is that general boundary conditions must be handled by manually tuned penalty weights or by carefully engineered trial spaces. The TENG-BC formulation directly contests that view: its discussion states that general boundary types are enforced intrinsically, avoiding penalty-weight tuning or trial-space construction (Jiang et al., 28 Feb 2026). Another misconception is that higher-order time integrators necessarily undermine the efficiency of neural solvers. The reported TENG++ trade-off is more nuanced: Heun requires roughly twice the PDE-residual evaluations per time step, but its superior temporal accuracy can allow larger $$\theta_{n+1}=\theta_n-\eta\,F(\theta_n)^{-1}\nabla_\theta L(\theta_n),$$28 for the same error tolerance, offsetting the extra cost (He et al., 13 Dec 2025).

In conceptual terms, TENG-BC is best understood as a localized time-stepping method in which the network supplies a continuous representation of the evolving field, while the Jacobian-induced metric supplies a geometry-aware correction at every step. The papers attribute its long-time behavior to local natural-gradient corrections that control stepwise error and suppress long-time drift, and its boundary robustness to a unified least-squares formulation that embeds the boundary operator directly into the update (Jiang et al., 28 Feb 2026).