Partial differential equations (PDEs) are fundamental to modeling complex phenomena across physics, chemistry, biology, and finance. Numerical solution of PDEs remains challenging due to grid discretization costs and the curse of dimensionality encountered by traditional methods such as finite element, finite difference, and finite volume approaches. Physics-informed neural networks (PINNs) provide a mesh-free alternative by embedding governing equations, boundary conditions, and initial conditions into a neural network loss via automatic differentiation. PINNs hold promise for direct and inverse problems, especially in data-scarce settings, but important limitations persist that reduce their accuracy and generalization.
Two interconnected limitations affect conventional PINNs. First, PINNs typically impose PDE constraints at a finite set of discrete collocation points, whereas the target equations are defined over continuous domains. This spatial mismatch can leave intervals between collocation points under-supervised and permit nonphysical oscillations in under-resolved regions. Second, learning difficulty is not uniform across the domain: boundary layers, discontinuities, and regions with rapid variation yield much higher residuals than smooth interior areas. Applying equal importance to all training samples induces an Unbalanced Prediction Problem (UPP), where optimization concentrates on easy regions, trapping the model in local minima and under-resolving critical complex structures.
Region optimization addresses the first limitation by expanding each collocation point into a continuous neighborhood. Instead of only enforcing the PDE at a single point, region optimization defines a region loss as an integral over a neighborhood of radius r around each sampling point. Because the region integral cannot be computed analytically, a Monte Carlo approximation is used: at each iteration, offsets are sampled uniformly from the neighborhood and the expected loss gradient at the sampled offsets approximates the region loss gradient. Taylor expansion arguments show that Monte Carlo sampling implicitly constrains the pointwise loss and its first-order derivatives, providing a higher-order regularization effect that suppresses unphysical oscillations in solutions with drastic variations.
To maintain stability, region optimization employs a trust-region calibration that adapts the region radius r based on gradient variance observed when sampling within the neighborhood. High gradient variance prompts shrinking r to preserve convergence stability, while low variance permits larger regions to enhance generalization. This dynamic calibration balances stronger local constraints against optimization stability.
Cognitive learning applies a curriculum-like progression in which the model dynamically adjusts emphasis on samples according to assessed difficulty. In the physics-informed setting used by CoPINN, difficulty for a sampling point is quantified as the Euclidean norm of the gradient of the PDE residual loss at that point. Larger gradient norms indicate greater difficulty in satisfying the PDE constraint and often correspond to boundary layers or rapid solution changes.
Cognitive weight allocation then evolves over training: early stages favor easy samples to establish a stable global structure; mid-training balances weights to avoid premature convergence and enable exploration of harder regions; late-stage training shifts emphasis toward hard samples to refine fitting and resolve residual local difficulties. This schedule mitigates the UPP by progressively reallocating optimization effort from easier to more difficult regions.
CoRoPINN (Cognitive Region Optimized PINNs) integrates region optimization and cognitive learning within a single closed feedback loop rather than treating them as independent modules. The loop couples a globally stable trust-region calibration with pointwise residual difficulty estimates. The same difficulty signal simultaneously adapts the sampling neighborhood for each collocation point and updates cognitive weights that modulate sample importance over training. By linking spatial continuity (region constraints) with temporal curriculum learning (cognitive weights), CoRoPINN focuses continuous supervision on under-resolved collocation points and progressively redirects optimization resources toward harder regions as training advances.
The principal mechanisms include a dual radius update—first identifying a globally stable region scale via gradient-stability statistics, then adapting each sampling neighborhood to local learning difficulty—and the joint use of residual-based difficulty to control both region sampling and cognitive weighting.
CoRoPINN implements region integrals via Monte Carlo approximation: at each iteration, offsets are sampled uniformly from the neighborhood around a collocation point and used to approximate the region loss gradient. Trust-region calibration computes gradient variance across sampled offsets to update a global stable radius; when variance is high, radius is reduced, and when variance is low radius can increase. Difficulty for individual samples is measured by the gradient norm of the PDE residual loss at each iteration. This difficulty informs both the radius adaptation for that point and the evolving cognitive weight used in the overall loss.
These interacting components produce an optimization trajectory that strengthens continuous constraints around hard points while progressively shifting training emphasis from easy to difficult regions, aiming to resolve local fine-scale structures and under-resolved dynamics.
CoRoPINN was evaluated on a set of benchmarks including three one-dimensional PDEs (1D-Reaction, 1D-Wave, and Convection) across five backbone architectures, Poisson problems in two, three, and five spatial dimensions, and Burgers2D. Across the five backbones, CoRoPINN achieved the lowest geometric-mean relative mean absolute error (rMAE) on all three one-dimensional benchmarks. Aggregate error reduction relative to the second-best method was reported as 21.0% for 1D-Reaction, 61.6% for 1D-Wave, and 52.9% for Convection. In Poisson benchmarks CoRoPINN ranked first in both metrics for 2D and 5D, and second in 3D. On Burgers2D it attained the lowest rMAE and relative root mean square error (rMSE) among the compared methods. Under the paper’s loss-ranking rule, CoRoPINN led or tied in 37 of 45 reported metric cells, demonstrating a consistent accuracy advantage within the evaluated scope.
Datasets for the 1D-Reaction, 1D-Wave, and Convection benchmarks are available at https://github.com/thuml/RoPINN. Datasets for the Poisson and Burgers2D benchmarks are available at https://github.com/i207M/PINNacle. The work received partial support from the National Natural Science Foundation of China under the grants reported in the source. The authors declared no competing interests.
CoRoPINN couples region-based continuous constraints with curriculum-style cognitive weighting in a closed feedback loop to address two key PINN limitations: insufficient supervision between collocation points and spatially uneven learning difficulty. By calibrating a global trust region, adapting pointwise neighborhoods according to residual difficulty, and progressively reallocating optimization toward harder regions, CoRoPINN improves accuracy across a range of PDE benchmarks and backbone architectures. The reported benchmark results indicate a clear overall accuracy advantage for CoRoPINN among the evaluated baselines.