engineering modelers

Abstract: Equation discovery solvers often use different data interfaces, equation representations and evaluation criteria, making their results difficult to compare. We present LegoSR, a unified framework for algebraic symbolic regression and differential equation discovery. LegoSR integrates seven solvers through a common equation representation and evaluation process, and supports searches constrained by partially known equations. Researchers can run solvers directly or use LegoSR Agent, which selects solvers and adapts searches based on solver knowledge, intermediate results and lessons from earlier tasks. On noisy ordinary and partial differential equation benchmarks, LegoSR Agent recovers nearly as many equations as the union of seven expert-tuned solvers and outperforms a generic agent with the same model, tools and budget by up to 29 percentage points. Three examples demonstrate basic use, adaptive solver selection and discovery guided by partial physical knowledge.

Abstract: In physics-informed machine learning, a target function $u^*$ is learned from noisy value observations $y_i=u^*(x_i)+ \varepsilon_i$, together with differential information, given either by noisy observations $d_j=(Du^*)(z_j)+ξ_j$ or by a known physical constraint $Du^*=v$. We consider the setting where $D$ is a linear differential operator and analyze a physics-informed kernel estimator $\hat u$ combining $n$ value observations and $m$ differential observations. In this context, we ask how much can differential information improve predictions, and how does this improvement depend quantitatively on $n$, $m$, and $D$. We prove finite-sample bounds, supported by numerical simulations, revealing a two-regime structure for the prediction error. When $m$ is limited, the rate depends jointly on $n$ and $m$; when $m$ exceeds a problem-dependent threshold, the rate saturates and matches the oracle rate obtained when the perfect constraint $D \hat u = Du^*$ is imposed. Examples are discussed for Sobolev spaces which are reproducing kernel Hilbert spaces and include partial Laplacian constraints on the torus and gradient observations on bounded domains. These examples illustrate the range of possible learning rate improvements --- from the standard nonparametric $n^{-1/4}$ to the parametric rate $n^{-1/2}$. Finally, we derive physically consistent rates in a stronger norm that jointly controls the errors in $\hat u$ and $D\hat u$.