World models of environment, agent and joint agent-environment systems

World models are a central component of model-based reinforcement learning. They are usually discussed in terms of what variables they predict, such as observations, rewards, states, latent or information states. We argue that there is a prior distinction: which channel they model. We consider three cases: the environment channel $O_{:} \mid A_{:}$, the agent channel $A_{:} \mid O_{:}$, and the realised joint process $(A, O)_{:}$, equivalently viewed as a channel with no inputs. Using computational mechanics, we define canonical predictive models for these three cases as $ε$-transducers or $ε$-machines. Canonical environment models recover standard predictive state representations, while the other two give analogous notions of canonical models for the agent and the joint system. We then build canonical support-restricted environment and agent models induced by closed-loop coupling, whose predictive equivalences range over continuations supported by the realised interaction. The key structural result is that canonical support-restricted environment states factor through the canonical joint causal states, and their transition structure is induced directly from the joint model; the agent-side construction is dual. Finally, we give a POMDP/controller example in which the unrestricted environment model has infinitely many states while the canonical support-restricted model induced by the coupling is finite. The framework clarifies what different world models are models of, and how coupling and support restriction can change their canonical predictive structure and complexity.

PG-KINN: A Physics-Informed Petrov-Galerkin Kolmogorov-Arnold Network for Solving Forward and Inverse PDEs

Physics-informed learning of partial differential equations (PDEs) has been dominated by multilayer perceptrons (MLPs), whose spectral bias and dense parameterization limit both accuracy and interpretability. Kolmogorov Arnold Networks (KANs) mitigate these limitations because their learnable spline activations are structurally aligned with the piecewise-polynomial bases of classical discretizations. However, the way a PDE is cast into a loss functional is as decisive as the choice of approximator: strong-form residual minimization requires high-order derivatives and heavily weighted losses, the energy (Bubnov-Galerkin) form is restricted to self-adjoint operators and, as we show, collapses to a trivial solution for parameter-identification problems, and boundary integral forms require a known fundamental solution. We propose PG-KINN, a physics-informed KAN built on a Petrov-Galerkin formulation in which the trial space is a KAN and the test space is an independent, compactly supported, piecewise-polynomial space evaluated with Gauss-Legendre quadrature. Integration by parts lowers the differentiation order while retaining applicability to general non-self-adjoint, nonlinear, and inverse problems; the localized test functions turn the global residual into a set of element-wise weak residuals with favorable conditioning. On a suite of benchmarks spanning crack singularities, stress concentration, Neo-Hookean hyperelasticity, inverse parameter identification in heterogeneous media, and complex geometries, PG-KINN consistently outperforms legacy MLP baselines and state-of-the-art KAN-based strong/energy/inverse formulations (PIKAN). These results position the Petrov-Galerkin coupling of KAN trial spaces and polynomial test spaces as a robust and accurate route for AI-based computational mechanics.

Physics-Audited Agentic Discovery in Scientific Machine Learning

In agentic scientific machine learning (SciML), large language model (LLM) agents can discover surrogate models and select one by an automated score, typically an error metric. A low error, however, does not establish that the predicted fields satisfy the physics that matter for mechanics, such as boundary conditions, superposition, stiffness scaling, or causality. We introduce Physics-Audited Agentic SciML (PA-SciML), a verification-first workflow for agentic SciML discovery. The workflow fixes a scoring evaluator before search, derives reviewable machine-checkable physics requirements, checks each trained candidate on its outputs, and separately searches prescribed input ranges or measured load-history spans for high-violation cases without reference solution fields. A surrogate is reported as verified only under the stated checks. When enabled, the workflow also adds advisory numerical probes before training and tests one modeling change at a time to record which isolated edits are associated with score gains before reuse. In the reported computational-solid-mechanics numerical examples, the static elasticity run selects a surrogate with lower validation error than the error-only baseline while both selected models pass the common linear-elastic checks. In the transient elastodynamics run, an error-only baseline with similar mean error fails a stricter causality check by responding to future parts of the loading history, while the selected surrogate passes the stated checks. The main distinction is per-candidate physics evidence on predicted fields, not a richer aggregate score.