A word that means "based on sound and not-arbitrary principles"
We propose a new well-founded method that does not suffer from the arbitrariness of parameter selection that is common in existing approaches.
well-founded (adj.)
Based on excellent reasoning, information, judgment, or grounds m-w
If you say that a report, opinion, or feeling is well-founded, you mean that it is based on facts and can therefore be justified. Collins
Having a foundation in fact; based on good reasons, information, etc. dictionary.com
Ecotoxicological studies, especially those concerned with biological monitoring and standardisation of..., are exceedingly difficult in the absence of a uniform, well-founded experimental procedure and quantitative treatment of the results obtained. S. A. Patin; Pollution and the Biological Resources of the Oceans
However, to the best of our knowledge no systematic ways, i.e. based on a well-founded methodology and guiding principles, for building geospatial ontologies have been proposed so far. C. Claramunt et al; GeoSpatial Semantics: 4th International Conference
And a poorly designed experiment leads to a weak scientific result—even the most sophisticated method of analysis cannot change that. On the other hand, a particularly well-qualified cook can still get "that certain something" out of a good recipe and a well-founded analytical method can derive an even more significant scientific insight from a well-design experiment. J. Weimann and J. Brosig-Koch; Methods in Experimental Economics
rigorous:
OED
3. Extremely detailed and thorough; precise, or concerned with precision; strictly accurate or exact.
1651 T. Hobbes Leviathan i. viii. 34 In Demonstration..and all rigourous Search of Truth, Judgement does all.
1954 H. Becker in H. Becker et al. For Sci. Social Man v. 147 It was so rigorous in method that it still serves as a reprimanding example to those who think that ‘description is easy’.
2003 G. J. Dorrien Making Amer. Liberal Theol. 1900–50 iv. 265 Wieman lectured that in the rigorous sense of the term, all knowledge is scientific.
The term grounded is well-suited for the purpose,
give (something abstract) a firm theoretical or practical basis
[Google's Oxford scraping (with apologies to the reference experts)]
with a little tweaking of your example sentence. A minor variation would be:
We propose a novel, well grounded method that does not suffer from the arbitrariness of parameter selection that is common in existing approaches.
The term's typical use requires a bit more massaging, but I think it accomplishes your goal.
We propose a novel method that has a well grounded theoretical basis, and does not suffer from the arbitrariness of parameter selection that is common in current approaches.
In the physics community, ab initio, Latin for "from the beginning", is widely used to describe research methods built on universal constants rather than arbitrary parameters. It's Latin, but perhaps it's ubiquitous enough to be considered a loanword to English.
For example: ab initio nuclear methods or ab initio quantum methods. A search for "ab initio" on the arXiv, the major preprint server used for physics papers, yields thousands of titles including "ab initio".
You might say
We propose a new ab initio method that does not suffer from the arbitrariness of parameter selection that is common in existing approaches.