OptPDE: Discovering Novel Integrable Systems via AI-Human Collaboration

TL;DR

Integrable PDEs are exceedingly rare and difficult to discover. The authors present OptPDE, an AI–human framework that optimizes PDE coefficients to maximize the number of conserved quantities $n_{ m CQ}$, guided by the CQFinder that symbolically extracts CQs. Through large-scale searches in a cubic-polynomial PDE basis, they identify four CQ-rich PDE families, including a novel one $u_t=(u_x+a^2u_{xxx})^3$; in the $a=0$ limit, the reduced equation $u_t=u_x^3$ exhibits an infinite hierarchy of CQs and intricate wave-breaking dynamics. This work demonstrates a promising AI–human collaboration loop where machine learning proposes interpretable hypotheses and humans verify and analyze them, enabling rapid discovery and interpretation of new integrable systems with potential physical relevance.

Abstract

Integrable partial differential equation (PDE) systems are of great interest in natural science, but are exceedingly rare and difficult to discover. To solve this, we introduce OptPDE, a first-of-its-kind machine learning approach that Optimizes PDEs' coefficients to maximize their number of conserved quantities, $n_{\rm CQ}$, and thus discover new integrable systems. We discover four families of integrable PDEs, one of which was previously known, and three of which have at least one conserved quantity but are new to the literature to the best of our knowledge. We investigate more deeply the properties of one of these novel PDE families, $u_t = (u_x+a^2u_{xxx})^3$. Our paper offers a promising schema of AI-human collaboration for integrable system discovery: machine learning generates interpretable hypotheses for possible integrable systems, which human scientists can verify and analyze, to truly close the discovery loop.

Figures (10)

• Figure 1: The OptPDE pipeline visualized. Given a basis of terms for a PDE, OptPDE optimizes the coefficients to maximize the number of conserved quantities (CQs) of the PDE, $n_{\rm CQ}$, the output of our CQFinder routine. Originally, $u$ decays and is not conserved, but OptPDE discovers coefficients that make $u$ more conserved by zeroing the diffusion term. The visualized example is trivial, but given an expansive PDE basis, OptPDE can help human scientists discover novel integrable systems. We also highlight which steps of our pipeline are done by humans and AI, creating a collaborative workflow where humans propose hypotheses and interpret results, while AI carries out tedious computations.
• Figure 2: CQFinder results for Burger's equation (left), the Korteweg–De Vries equation (center) and the Non Linear Schrödinger equation (right) parameterized in real ($u$) and imaginary ($v$) parts. CQFinder correctly returns the number of CQs ($n_{\rm CQ}$) for each of these cases, in addition to their symbolic formulas (Appendix \ref{['symbolic_forms']}), and identifies trivial solutions that have a simple antiderivative. The ability for CQFinder to correctly identity $n_{\rm CQ}$ for PDEs lays the foundation for optimizing general PDE forms with OptPDE.
• Figure 3: Each point on this 3D PCA corresponds to a PDE that OptPDE returns given PDE and CQ bases. We discover four families of PDEs with at least one conserved quantity, listed above. Note that these PDE families form bases for the subspace of OptPDE's results. For each family of PDEs, we use the transformation $x = ax'$ such that $u_{nx'} = a^nu_{nx}$ and plot the result in our PCA space for varying values of $a$. The $a = -1,1$ cases are shown with larger marker sizes.
• Figure 4: Human analysis of an AI-discovered integrable system. Left: We evolve a Gaussian (top) and sine wave (bottom) according to the equation $u_t = u_x^3$, which is a reduced form of the integrable PDE $u_t=(u_x+a^2u_{xxx})^3$ OptPDE discovered. There is a time at which a "break" forms and both curves becomes non differentiable at a point. This break passes through the curves until they degrade into what visually appears to be straight lines. Right: A phenomenological model to explain the magnitude decay of the equation $u_t=u_x^3$ after breaking happens.
• Figure 5: CQFinder results for Burger's equation (left), the Korteweg–De Vries equation (center) and the Non Linear Schrödinger equation (right) parameterized in real ($u$) and imaginary ($v$) parts. Given a user-inputted CQ basis, CQFinder uses linear regression to determine the conserved quantities of a PDE using a singular value threshold of $10^{-4}$. CQFinder also computes which of these conserved quantities are a trivial antiderivative of a simple function. We rediscover all expected conserved quantities for these examples. Note that the human scientist is still required for the interpretation of these results. For example, CQ3 for KdV is listed as $0.94u^3+0.24u_x^2-0.24u_{xx}u$. The human scientist must realize that you can add any function with a symbolic antiderivative to a CQ, note that $u_x^2+uu_x = \frac{d}{dx}uu_x$, so CQ2 is $0.94u^3+0.48u_x^2$. This closely matches KdV's third integral of motion, $2u^3+u_x^2$, but it is once again up to the human scientist to look past the small difference in coefficients and identify them as the same.