Agentic Symbolic Search: Characterizing PDEs Beyond Hand-crafted Expressions, Meshes, and Neural Networks

Abstract: Mathematicians understand a PDE solution through mathematical structures rather than tables of computed values. Historically, this has been the product of mathematical analysis, carried out by hand for each problem individually. Neither numerical simulation nor neural networks produce those structures directly. We propose Agentic Symbolic Search (ASYS), a prior-guided framework in which an agent translates PDE theory, public problem constraints, and accumulated search experience into testable differentiable symbolic programs. The mathematical forms are refined under evolutionary search, while their continuous parameters are fit by gradient-based optimization. This makes the search an automated form of inductive-bias injection rather than blind symbolic regression. For problems with known analytical forms, ASYS recovers these forms naturally; for other problems, ASYS constructs analytical approximations which can guide mathematicians toward further analysis. In our experiments, across five problems spanning bounded dynamics, finite-time blow-up, and free-boundary focusing, ASYS produces interpretable representations, including a geometric interface formula for Allen-Cahn 2D dynamics and a nine-parameter contraction law for Keller-Segel chemotactic blow-up, in settings where no closed-form description was previously available. ASYS shows the possibility of a new paradigm for characterizing PDE solutions, beyond handcrafted analytical solutions, mesh-based numerical solutions, and neural network approximations.

• The paper introduces an agent-driven method that automates the generation of explicit, symbolic representations for PDE solutions.
• It combines a two-loop framework—structural search and parameter optimization—to accurately recover interpretable forms for bounded, singular, and free-boundary cases.
• The approach outperforms traditional neural surrogates in terms of interpretability and efficiency, while also highlighting challenges in capturing global similarity in high-dimensional problems.

Agentic Symbolic Search for Symbolic Characterization of PDEs

Context and Motivation

Partial differential equations (PDEs) are fundamental for describing physical systems' dynamics. Traditional approaches revolve around hand-crafted analytical solutions, mesh-based numerical methods, or more recently, neural network surrogates. Analytical solutions are prized for their explicit structure, interpretability, and insight into asymptotic regimes, but the methodology is inherently ad hoc and only viable for select cases. Mesh-based solvers generalize the scope but yield non-interpretable tables of values, while neural surrogates (e.g., PINNs, neural operators) improve scalability in high dimensions but lock solution structure within weights, impairing explicit interpretability and limiting expressive power for singular or structurally intricate phenomena.

Agentic Symbolic Search (ASYS) advances a paradigm shift: the explicit mathematical structure of PDE solutions can be generated via automated search rather than exclusively by human mathematical analysis. ASYS operationalizes this by leveraging coding agents informed by mathematical prior and problem constraints, iteratively generating and optimizing differentiable symbolic programs in search of interpretable solution representations. This addresses not merely parameter fitting but structural adaptation—searching over forms, not just coefficients—under guidance from physical insight and literature.

Methodological Framework

ASYS accepts a concrete PDE, initial and boundary conditions, and relevant constants as input. The search proceeds through two strictly decoupled nested loops:

• Outer Loop (Structural Search): Guided by problem specification, mathematical priors, previous high-scoring candidates, and diagnostic feedback, an agent proposes revised program structures (the ansatz)—including coordinate transforms, branch decompositions, and trainable parameterizations. The hypothesis space spans differentiable, closed-form symbolic programs rather than unrestricted code or simple expression trees as in symbolic regression.
• Inner Loop (Parameter Optimization): For each structural candidate, continuous parameters are fitted using quasi-Newton L-BFGS under a fixed computational budget, operating exclusively on agent-engineered scalar losses.

Scoring relies on a four-component vector capturing physics residual, initial value accuracy, boundary consistency, and a compatibility condition ensuring correct temporal departure from the initial manifold—strictly using public constraints and never solution reference data. Selection is managed by an Evolutionary Ensemble of Agents (EvE), gradually biasing toward structurally and numerically superior representations.

Empirical Results

ASYS is evaluated across five PDEs, encompassing bounded dynamics, finite-time blow-up, and free-boundary singularities. Experiments are executed on CPU-only hardware, emphasizing method efficiency and interpretability over neural surrogacy.

Bounded Cases

Nonlinear Schrödinger Equation (NLS):

ASYS reconstructs the exact Satsuma–Yajima breather profile, augmenting it with minimal endpoint corrections for periodicity. The solution is a mix of analytic soliton form and five trainable scalars, achieving a validation relative $L_2$ error of 0.0059, confirming fully interpretable, symbolic recovery.

Allen–Cahn 2D Equation:

For a geometric peanut-to-oval merger, ASYS constructs a signed-distance-based geometric scaffold with explicit time-dependent blending between peanut and oval regimes and localized neck corrections. 23 interpretable scalars encode shrink rates, oval axes, neck dynamics, and memory decay. Achieved validation relative $L_2$ error is 0.0107, confirming accurate, fully symbolic geometric tracking.

Singular and Free-Boundary Cases

Keller–Segel Radial Blow-up:

Without a priori global similarity coordinates, ASYS discovers the critical contraction law underpinning finite-time blow-up with nine scalars, outperforming a hand-supplied self-similar PINN baseline (ASYS: $L_2$ 0.188 vs SS-PINN: 0.258). This representation encapsulates core mass dynamics, halo separation for supercritical excess mass, and explicit contraction scaling, with minimal parameterization (nine scalars vs 49,921 MLP weights).

Graveleau Porous Medium Focusing:

ASYS structurally instantiates a classical second-kind self-similar regime for PME focusing, learning the nonlinear similarity exponent $\beta$ indirectly through residual minimization ($\beta_{\text{model}} = 0.928$, reference $\beta_{\text{ref}} = 0.877$) and matching pressure profiles to a relative $L_2$ error of 0.00132. Structural alignment rather than local residual reduction is shown to drive optimal solution accuracy.

gCLM Stress Test:

On the generalized Constantin–Lax–Majda equation with strong nonlocal coupling, ASYS identifies its expressive boundary: despite repeated ansatz refinement and explicit Hilbert transform inclusion, it fails to construct the global similarity coordinate within the search horizon. The best candidate (Taylor-anchored neural ansatz) achieves only $L_2$ error of 0.465, compared to SS-PINN at 0.196, isolating structural flexibility as the limiting bottleneck.

Implications and Theoretical Remarks

ASYS demonstrates that interpretable, high-fidelity symbolic representations can be constructed for individual, structurally challenging PDE evolutions with modest computational resources, circumventing traditional hand analysis and neural network opacity. Its explicit structure—coordinate transformations, branching, geometric mechanisms—directly exposes scale, similarity, and singularity regimes validated against independent numerical solutions. This is in sharp contrast with neural surrogates, which trade interpretability for amortized inference; ASYS trades amortization for structural insight.

The method's limitations center on agentic structural reach, not scoring—failures to discover global similarity frameworks (as in gCLM) reflect the difficulty of exhaustive symbolic exploration for strongly nonlocal or high-dimensional phenomena. Scoring, abstracted from observed trajectory data, enables generalization to forward, inverse, and partially specified problems. Future developments may combine agentic symbolic cores with neural or spectral surrogates for residual fields and employ complementary optimization strategies to mitigate non-convex fitting landscapes. ASYS-produced representations could serve as seeds for conjecture, rigorous analysis, or verification in mathematical PDE theory.

Conclusion

Agentic Symbolic Search provides a computational mechanism for automatic structural characterization of PDE solutions, bridging the gap between mesh-based and neural-network approaches. It recovers explicit analytical and geometric forms for bounded, singular, and free-boundary problems, produces interpretable contractions and scaling laws where closed forms are absent, and quantitatively matches or surpasses neural surrogates when structural priors are lacking. The framework is poised to extend to inverse problems and real-world systems as agentic search and optimization capabilities mature, marking an evolution in automating mathematical discovery and interpretability within computational science.