Cognitive and Physics-Informed Methods

• Cognitive and physics-informed methods are defined as frameworks that integrate governing physical models with data-driven learning to enforce consistency and improve simulation fidelity.
• These approaches utilize residual-based learning, kernel methods, and graph networks, employing techniques like automatic differentiation and latent state modeling for enhanced stability.
• They are applied across diverse domains such as fluid mechanics, biomedical imaging, climate forecasting, and robotics, offering robust performance in forward modeling and inverse design.

Cognitive and physics-informed methods are learning frameworks in which mathematical models of the world are combined with data, and in which the learning system is often described as “cognitive” when it reasons with prior knowledge, maintains internal state, supports hypothesis testing, or uses internal models for sequential prediction and control. In scientific machine learning, physics-informed machine learning involves the use of neural networks, graph networks or Gaussian process regression to simulate physical and biomedical systems, using a combination of mathematical models and multi-modality data (Shukla et al., 2022). In adjacent reviews, physics enters through synthetic data generation, non-dimensionalization and scaling, architecture selection, loss design, and post-training regularization, while computer-vision formulations distinguish modification of input data, modification of network architectures, and modification of training losses as the main routes by which governing physical equations are incorporated (Labeb et al., 20 Jun 2026, Banerjee et al., 2023).

1. Foundations and scope

Physics-informed learning is commonly formulated as constrained optimization. A representative PINN formulation considers a PDE system

$$\mathcal{N}(u(x,t;\lambda)) = 0 \quad \text{in } \Omega\times(0,T),$$

with boundary conditions $\mathcal{B}(u)=g$ and initial condition $u(x,0)=h(x)$, and trains a neural approximation $u_\theta(x,t)$ with a composite objective

$$\mathcal{L}(\theta,\lambda) = \lambda_d\,\mathcal{L}_{\text{data}} + \lambda_f\,\mathcal{L}_{\text{physics}} + \lambda_b\,\mathcal{L}_{\text{BC/IC}},$$

where the physics residual is evaluated by automatic differentiation (Labeb et al., 20 Jun 2026). In broader scientific machine learning, the same pattern appears in regularized empirical risk minimization, kernel methods, operator learning, graph networks, and model-based reinforcement learning.

The “cognitive” designation is used in several distinct but related senses. One line emphasizes strong structured priors: physical laws restrict the admissible hypothesis space, much as intuitive physics restricts plausible world-models (Scampicchio et al., 29 Sep 2025). Another line emphasizes sequential reasoning: models maintain memory, predict future states from past states, and enforce consistency of rollouts with known laws (Nagda et al., 22 Aug 2025). A third line emphasizes internal models for planning and control: in physics-informed model-based reinforcement learning, a learned dynamics model generates imaginary trajectories and supports analytic policy gradients (Ramesh et al., 2022). A plausible implication is that the term “cognitive” in this literature is less a claim about human-level cognition than a statement about structured, prior-guided inference.

The scope of these methods is correspondingly broad. Reviews cover forward modelling, inverse design, and equation discovery across nanophotonics, fluid mechanics, astronomy, and biomedical engineering (Labeb et al., 20 Jun 2026). Domain-specific studies instantiate the same principles in brain MRI segmentation (Borges et al., 2020), optical modes in composites (Ghosh et al., 2021), global weather forecasting (Nagda et al., 22 Aug 2025), auditory fMRI decoding (Ma et al., 2024), climate prediction on graphs (Seo et al., 2019), and parametric PDE solution (Boudec et al., 2024).

2. Modes of incorporating physics

The most familiar mechanism is residual-based learning. In canonical PINNs, the PDE, initial conditions, and boundary conditions are penalized at collocation points, and derivatives are computed with automatic differentiation (Nagda et al., 22 Aug 2025). Physics-informed kernel learning replaces nonconvex network training with a physics-informed empirical risk on Sobolev spaces,

$$R_n(f) = \frac{1}{n}\sum_{i=1}^n |f(X_i) - Y_i|^2 + \lambda_n \|f\|^2_{H^s(\Omega)} + \mu_n \|\mathscr{D}(f)\|^2_{L^2(\Omega)},$$

where the third term penalizes violation of a known linear differential operator (Doumèche et al., 2024). In the dependent-data setting, an analogous PDE regularizer

$$R(f) := \|D(f)\|_{L^2(\mathcal{X}^T;\mathbb{R}^{d_Y})}^2$$

shrinks learning toward functions that satisfy the known PDE and thereby reduces effective complexity (Scampicchio et al., 29 Sep 2025).

Not all physics-informed methods use residual penalties in the same way. In brain MRI segmentation, the method is physics-informed through forward modeling and conditioning rather than a PINN-style physical loss: large synthetic datasets are generated from static MR signal equations, and a 3D U-Net receives both the image and acquisition parameters such as $[TI,e^{-TI}]$ or $[TR,TE,FA,e^{-TR},e^{-TE},\sin(FA)]$ (Borges et al., 2020). In optical modes in composites, the physics-informed loss is a normalized residual of the Maxwell eigenvalue equation,

$$L_P = \frac{\sum_m\left|\sum_j \hat A_{mj}h_j - k_y^2 h_m\right|^2} {\sum_m \left|\sum_j \hat A_{mj}h_j \right|^2},$$

and unlabeled configurations contribute through this residual even when no ground-truth eigenmodes are available (Ghosh et al., 2021).

Graph-based formulations encode physics through discrete differential structure. Physics-informed graph networks use graph exterior calculus, with operators such as

$\mathcal{B}(u)=g$0

and Hodge Laplacians to represent conservation laws and graph PDEs (Shukla et al., 2022). Differentiable Physics-informed Graph Networks instead regularize latent graph dynamics directly, for example with a diffusion-style constraint

$\mathcal{B}(u)=g$1

so that latent states evolve according to a soft physical prior (Seo et al., 2019). In computer vision more generally, this diversity is summarized as observation bias, inductive bias, and learning bias (Banerjee et al., 2023).

3. Cognitive motifs: memory, internal models, and sequential reasoning

Autoregressive dynamics are the clearest site where cognitive language becomes technical. PIANO is proposed to align the neural architecture with the autoregressive nature of dynamical systems. Instead of learning a global mapping $\mathcal{B}(u)=g$2, it learns a state transition

$\mathcal{B}(u)=g$3

with hidden state updates

$\mathcal{B}(u)=g$4

and a PDE probe that outputs the next state (Nagda et al., 22 Aug 2025). The paper explicitly characterizes this as memory of past states, autoregressive prediction, sequential reasoning under constraints, and self-supervised rollout / imagination. This suggests a concrete bridge between sequence modeling and physics-informed learning: physical consistency is enforced on trajectories generated from the model’s own internal state.

Model-based reinforcement learning offers a second formulation of internal world models. In the rigid-body setting, the state is $\mathcal{B}(u)=g$5 and the dynamics obey

$\mathcal{B}(u)=g$6

A physics-informed Lagrangian Neural Network learns potential energy $\mathcal{B}(u)=g$7 and a lower-triangular factor $\mathcal{B}(u)=g$8 such that $\mathcal{B}(u)=g$9, computes accelerations from the Euler–Lagrange structure, and is then differentiated through short-horizon imaginary rollouts to optimize a policy (Chen et al., 2022). The paper argues that model accuracy mainly matters in environments that are sensitive to initial conditions, where numerical errors accumulate fast, and reports that the physics-informed version achieves significantly better average-return and sample efficiency in those environments.

Cognitive language also appears outside dynamical forecasting. In auditory fMRI decoding, PEMT-Net builds neural embedding graphs with random walks to simulate the physical diffusion of neural information and adds a position encoding based on relative physical coordinates (Ma et al., 2024). In semi-supervised PINN and PIGP training, label propagation promotes high-confidence predictions at unlabeled points to pseudo-labeled data, and the paper explicitly frames this as iterative propagation and refinement of information forward in time (Zhong et al., 2024). A plausible implication is that “cognitive” in this literature often denotes iterative internal inference under structural constraints rather than any single architectural family.

4. Architectural families and application domains

One large architectural family consists of operator-learning methods. Neural operators learn maps between function spaces rather than pointwise regressions, with the generic objective

$u(x,0)=h(x)$0

and Fourier Neural Operators implement nonlocal kernels through truncated Fourier modes (Viswanath et al., 2023). Physics-Informed Neural Operators augment supervised operator learning with PDE and boundary residuals, for example

$u(x,0)=h(x)$1

for stationary problems (Viswanath et al., 2023). Physics-Informed Koopman Networks occupy a related operator-learning niche: they learn encoder coordinates that approximate Koopman eigenfunctions while enforcing the Lie-operator constraint

$u(x,0)=h(x)$2

thereby reducing the need of large training data-sets while maintaining high effectiveness in approximating Koopman eigenfunctions (Liu et al., 2022).

Another family couples physics to recurrent, graph, or state-space architectures. PIANO uses an autoregressive state-space model over time and reports state-of-the-art performance on challenging time-dependent PDEs and weather forecasting (Nagda et al., 22 Aug 2025). Physics-informed graph networks use graph exterior calculus to construct differential operators on unstructured graphs, and DPGN injects a diffusion prior into latent graph dynamics to improve multistep climate prediction and inductive learning across regions (Shukla et al., 2022, Seo et al., 2019). Physics-informed machine learning for optical modes uses a custom regression layer to combine eigenvalue prediction, modal overlap, and Maxwell residuals in an ANN with fully connected layers and LSTM modules (Ghosh et al., 2021).

A third family uses forward models and parameter-aware architectures in sensing and imaging. Physics-informed brain MRI segmentation combines multiparametric MRI-based static-equation sequence simulations with a 3D U-Net that receives both the image and acquisition parameters, improving robustness to acquisition variations and helping bridge multi-centre and longitudinal imaging studies (Borges et al., 2020). Physics-informed computer vision reviews a broad landscape in which optical, mechanical, and sensor physics modify input data, architectures, or losses in tasks ranging from lensless imaging to human motion and neural radiance fields (Banerjee et al., 2023).

These applications are heterogeneous but structurally similar. Each introduces a representation—latent coordinates, graph states, operator features, or parameter-conditioned embeddings—in which known physics is easier to express. This suggests that the unifying design pattern is not a single loss or architecture, but the construction of a representation where physical law acts as a strong inductive bias.

5. Stability, learning rates, and empirical behavior

A major theoretical theme is that physical structure can alter not only accuracy but the learning regime itself. In dependent-data regression, aligned physics-informed regularization improves the learning rate from the Sobolev minimax rate to the optimal i.i.d. one without any sample-size deflation due to data dependence (Scampicchio et al., 29 Sep 2025). The same paper identifies the effective hypothesis class

$u(x,0)=h(x)$3

and shows that when the physical prior is aligned so that $u(x,0)=h(x)$4, the fast $u(x,0)=h(x)$5 term dominates the excess-risk bound (Scampicchio et al., 29 Sep 2025). Physics-informed kernel learning reaches a related conclusion through effective dimension analysis and reports that PIKL can outperform physics-informed neural networks in terms of both accuracy and computation time, while also identifying cases where PIKL surpasses traditional PDE solvers, particularly in scenarios with noisy boundary conditions (Doumèche et al., 2024).

Temporal stability is another recurring theme. PIANO formalizes one-step rollout inconsistency for non-autoregressive PINNs through

$u(x,0)=h(x)$6

and proves an error propagation bound

$u(x,0)=h(x)$7

arguing that standard PINN losses do not constrain $u(x,0)=h(x)$8 and therefore permit temporal instability, while autoregressive modeling and rollout-based training in PIANO control this source of error (Nagda et al., 22 Aug 2025). In label-propagation training for PINNs and Gaussian processes, iterative pseudo-labeling ameliorates the issue of propagating information forward in time, which is a common failure mode of physics-informed machine learning (Zhong et al., 2024).

Empirical results across domains reinforce these theoretical patterns. In optical modes, physics-informed learning dramatically improves accuracy and generalizability, and unlabeled configurations become useful because the Maxwell residual can be evaluated without ground-truth eigenmodes (Ghosh et al., 2021). In model-based robotics, the physics-informed version achieves better average-return than Soft Actor-Critic in challenging environments because it computes the policy-gradient analytically, while the latter estimates it through sampling (Chen et al., 2022). In climate prediction, DPGN outperforms GN-only, GN-skip, LSTM, and MLP baselines in one-step and multistep prediction, and the diffusion equation in latent space yields stable state transition and slowly varying latent states (Seo et al., 2019).

6. Limitations, controversies, and emerging directions

The literature is explicit that correct priors help and incorrect priors can hurt. In the dependent-data theory, if $u(x,0)=h(x)$9, the slow term does not vanish and misaligned physics can hurt by introducing bias (Scampicchio et al., 29 Sep 2025). Physics-informed model-based reinforcement learning likewise notes that strong priors are a double-edged sword: they are beneficial when roughly correct and harmful when far off (Chen et al., 2022). In kernel learning, current guarantees and efficient constructions are developed for linear constant-coefficient operators and low-dimensional settings, with Fourier truncation suffering from the curse of dimensionality (Doumèche et al., 2024).

Optimization remains a central controversy. Reviews of PINNs emphasize non-convex landscapes, spectral bias, and loss stiffness, especially in multi-scale and discontinuous regimes (Labeb et al., 20 Jun 2026). A dedicated survey argues that PINN loss landscapes are uniquely complex and identifies gradient-free evolutionary algorithms, hybrid gradient–evolution schemes, multi-objective optimization, and meta-learning of generalizable PINN models as important avenues for future research (Wong et al., 11 Jan 2025). A separate line proposes learning a neural solver for parametric PDEs so that a network conditions gradient descent on the PDE instance,

$u_\theta(x,t)$0

thereby accelerating and stabilizing optimization across a distribution of PDE parameters (Boudec et al., 2024).

Scalability is equally unresolved. Reviews identify the need to scale PINNs, PIGNs, and more broadly GNNs for large-scale engineering problems, with domain decomposition, mixed precision, and distributed graph frameworks as major algorithmic directions (Shukla et al., 2022). PIANO notes the cost of backpropagation through time for very long horizons and very high-dimensional systems (Nagda et al., 22 Aug 2025). MRI and auditory-fMRI studies note that their physics is simplified: static signal equations rather than full Bloch simulation in one case, and random-walk diffusion and Fruchterman–Reingold layouts rather than full neural dynamics in the other (Borges et al., 2020, Ma et al., 2024). This suggests that one of the main frontier questions is not whether physics should be injected, but at what level of fidelity and abstraction it should be injected so that the prior is strong enough to regularize and weak enough to remain useful under model misspecification.

Taken together, cognitive and physics-informed methods form a research program in which learning is guided by governing law, relational structure, and internal state. Across PINNs, neural operators, graph networks, Koopman models, kernel methods, and model-based control, the common ambition is to replace unconstrained function fitting with representations that are physically plausible, data-efficient, and stable under sequential use.