Mod­el­ling dy­nam­ic­ally evolving phys­ical sys­tems is at the core of sci­ence and tech­no­logy. En­gin­eers need to know how the wings of a new air­plane model will vi­brate un­der par­tic­u­lar flight con­di­tions, and cli­mate sci­ent­ists are try­ing to pre­dict how global tem­per­at­ures and weather pat­terns will de­velop in the fu­ture.  Those tasks are dif­fi­cult be­cause the un­der­ly­ing sys­tems, by their very nature, do not be­have lin­early. This means that, for in­stance, ap­ply­ing twice as much force to an air­plane wing will not ne­ces­sar­ily cause it to bend twice as far (it could bend either more or less than that).

Sci­ent­ists man­age to model such non­lin­ear dy­nam­ical sys­tems either by mak­ing lin­ear ap­prox­im­a­tions or by as­sum­ing spe­cific non­lin­ear equa­tions and then match­ing the model to ob­served data. How­ever, both ap­proaches lead to mod­els that are of­ten only valid over a lim­ited range of the sys­tem’s mo­tions. A group of sci­ent­ists led by George Haller, Pro­fessor of Non­lin­ear Dy­nam­ics at ETH Zurich, to­gether with re­search­ers at the Uni­ver­sity of Bre­men, have now found a new way to get com­puters to ex­tract, dir­ectly from ex­per­i­mental data, non­lin­ear dy­nam­ical mod­els that can make sub­stan­tially more ac­cur­ate pre­dic­tions than pre­vi­ous al­gorithms.

The lim­its of static ma­chine learn­ing

In re­cent years, re­search­ers have made tre­mend­ous pro­gress in teach­ing com­puters how to re­cog­nize pat­terns, faces and even hu­man speech. “Those are in­cred­ible achieve­ments”, says Haller, “but such ma­chine learn­ing ap­proaches are de­signed for fun­da­ment­ally static prob­lems. By con­trast, get­ting com­puters to learn the be­ha­viour of dy­nam­ical sys­tems, even of ap­par­ently simple ones like wa­ter slosh­ing in a tank, is sig­ni­fic­antly harder.”  A com­plete phys­ical model for slosh­ing wa­ter would have to in­clude not just the bulk flow of fluid, but also other phe­nom­ena, such as waves break­ing on the sur­face. Con­ven­tional sim­u­la­tions that in­clude all those fea­tures are ex­tremely time con­sum­ing even on mod­ern su­per­com­puters.

“Our new ap­proach re­lies on the real­iz­a­tion that one doesn’t need to re­pro­duce all the de­tails of the dy­nam­ics, but only its key struc­tures”, says Mat­tia Cene­dese, a postdoc in Haller’s group and first au­thor of the study just pub­lished in the sci­entific journal Nature Com­mu­nic­a­tionscall_made.

Get­ting the big pic­ture

Tak­ing the ana­logy of fa­cial re­cog­ni­tion, rather than con­sid­er­ing the de­tails of a hu­man face down to tiny wrinkles or even in­di­vidual pores in the skin, the com­puter al­gorithm de­veloped by the ETH re­search­ers looks the big pic­ture – say, the gen­eral shape of the eyes and nose. Ap­plied to dy­nam­ical sys­tems, this cor­res­ponds to find­ing com­bin­a­tions, for in­stance, of the po­s­i­tion and ve­lo­city of a part of the sys­tem rather than par­tic­u­lar tra­ject­or­ies un­der spe­cific cir­cum­stances. As a res­ult, the time needed for the cal­cu­la­tions can be re­duced from sev­eral hours or even days to just a few minutes.

To demon­strate the strength of their al­gorithm, Haller and his co-​workers used the res­ults of a water-​tank ex­per­i­ment per­formed by their Ger­man col­leagues. In that ex­per­i­ment, a trans­par­ent tank filled with wa­ter was ini­tially shaken back and forth un­til the wa­ter star­ted slosh­ing peri­od­ic­ally. The shak­ing of the tank was sud­denly stopped, and the wa­ter was filmed as the slosh­ing slowly sub­sided. From that foot­age the mo­tion of the centre of mass of the wa­ter was cal­cu­lated and fed into a com­puter. The al­gorithm then pro­duced a simple but still non­lin­ear math­em­at­ical model that cap­tured the ob­served slosh­ing mo­tion with high ac­cur­acy.

Journal: Nature
DOI: 10.1038/s41467-​022-28518-y 
Article Title: Data-driven modeling and prediction of non-linearizable dynamics via spectral submanifolds
Article Publication Date: 15-Feb-2022